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  • Unknown's avatar

    Jim Randell 8:33 am on 30 January 2026 Permalink | Reply
    Tags:   

    Teaser 2331: Everyone’s invited 

    From The Sunday Times, 27th May 2007 [link]

    The letters A, B, C, D, E, F, G, H and I stand for the digits 1 to 9 in some order. Thus AB, CD and EF are two-digit numbers; [and] GHI is a three-digit number. George and Martha arranged a big wedding for their granddaughter. The top table consisted of AB members (fewer than 60) of their extended family. There were CD other tables each seating EF guests. The total attendance was GHI (fewer than 500).

    What was the total attendance?

    [teaser2331]

     
    • Jim Randell's avatar

      Jim Randell 8:34 am on 30 January 2026 Permalink | Reply

      Here is a solution using the [[ SubstitutedExpression ]] solver from the enigma.py library.

      It runs in 92ms. (Internal runtime of the generated code is 14.8ms).

      #! python3 -m enigma -rr

SubstitutedExpression

--digits="1-9"

"A < 6"  # or: "AB < 60"
"G < 5"  # or: "GHI < 500"
"AB + CD * EF = GHI"

--template="(AB CD EF GHI)"
--answer="GHI"

      Solution: The total attendance was 489.

      The top table seated 57.

      And then there were either 12 additional tables, each seating 36; or 36 additional tables, each seating 12.

      The total attendance is thus: 57 + 12 × 36 = 489.

      Like

    • ruudvanderham's avatar

      ruudvanderham 9:46 am on 30 January 2026 Permalink | Reply 

      import peek
import istr

for A, B, C, D, E, F, G, H, I in istr.permutations(range(1, 10)):
    if A <= 5 and G <= 4 and (A | B) + (C | D) * (E | F) == (G | H | I):
        peek((A | B), (C | D), (E | F), (G | H | I))

      Like

    • GeoffR's avatar

      GeoffR 5:52 pm on 30 January 2026 Permalink | Reply 

      from itertools import permutations
dgts = set('123456789')

for p1 in permutations(dgts, 2):
    A, B = p1
    AB = int(A + B)
    if AB < 60:
        q1 = dgts.difference({A, B})
        for p2 in permutations(q1):
            C, D, E, F, G, H, I = p2
            CD, EF = int(C + D), int(E + F)
            GHI = int(G + H + I)
            if GHI == AB + CD * EF:
                if GHI < 500:
                    print(f"Total attendance = {GHI}")

# Total attendance = 489

      Like

    • BRG's avatar

      BRG 9:30 am on 3 February 2026 Permalink | Reply 

      # 10.A + B + (10.C + D).(10.E + F) == 100.G + 10.H + I < 500
# note: the order of C and E is undetermined (use C < E)

from itertools import permutations

# start with C and E: 100.C.E < 500  E < 4 // C + 1
for C in range(1, 10):
  for E in range(C + 1, 4 // C + 1):
    s8 = set(range(1, 10)) - {C, E}
    # complete the number of tables (CD) and the number on tables (EF)
    for D, F in permutations(s8, 2):
      CD, EF = 10 * C + D, 10 * E + F
      s6 = s8 - {D, F}
      # now find the number on the top table (AB < 60)
      for A, B in permutations(s6, 2):
        if (AB := 10 * A + B) < 60:
          # find the attendance (< 500) and ensure correct digit allocations
          GHI = (AB := 10 * A + B) + CD * EF
          s_ghi = {int(x) for x in str(GHI)}
          if GHI < 500 and len(s_ghi) == 3 and s_ghi.issubset(s6 - {A, B}):
            print(f"({AB = }) + ({CD = }) x ({EF = }) == ({GHI = })")

      Like

  • Unknown's avatar

    Jim Randell 10:05 am on 27 January 2026 Permalink | Reply
    Tags: by: Derek Emley   

    Teaser 2415: [Double shift] 

    From The Sunday Times, 4th January 2009 [link]

    I have written down a large number in which no digit occurs more than twice. If I were to move the last digit of the number from the end to the front of the number, then the effect would be to double the number. If I repeated the process with the new number, then the effect would be to double it again. And if I repeated the process again, the effect would be to double the number yet again.

    What is the number that I have written down?

    This puzzle was originally published with no title.

    [teaser2415]

     
    • Jim Randell's avatar

      Jim Randell 10:06 am on 27 January 2026 Permalink | Reply

      If we suppose the original number n is of the form:

      n = 10a + z
      where z is the final digit, and a is a k-digit number

      Then moving the final digit to the front gives us:

      2n = (10^k)z + a

      From which we derive that a is a k-digit number, such that:

      19a = (10^k − 2)z

      If the number contains no digit more than twice, then the longest the number can be is 20 digits long, and so we can consider k up to 19.

      This Python program runs in 70ms. (Internal runtime is 719µs).

      from enigma import (irange, div, nsplit, rotate, zip_eq, seq2str, printf)

# check a candidate number <n> (with digits <ds>)
def check(n, ds):
  # check no digit occurs more than twice
  if any(ds.count(x) > 2 for x in set(ds)): return
  # record shifts that result in doublings
  ns = [n]
  # collect doubling sequences
  while True:
    n *= 2
    ds = rotate(ds, -1)
    if not zip_eq(ds, nsplit(n)): break
    ns.append(n)
  return ns

# look for up to 19 digits prefixes
for k in irange(1, 19):
  x = 10**k - 2
  # consider possible final digits
  for z in irange(1, 9):
    a = div(x * z, 19)
    if a is None: continue
    n = 10*a + z
    ds = nsplit(n)
    if len(ds) != k + 1: continue
    ns = check(n, ds)
    if ns and len(ns) >= 4:
      printf("{ns}", ns=seq2str(ns, sep=" -> ", enc=""))

      Solution: The number is: 105263157894736842.

      The number is 18 digits long, and uses each digit twice except 0 and 9 (which are used once each).

      Shifting the final digit to the front multiple times we get:

      105263157894736842 = n
      210526315789473684 = n × 2
      421052631578947368 = n × 4
      842105263157894736 = n × 8

      A run of 4 is the longest run we can get in base 10 (as n × 16 will not have the same number of digits as n), but there is another run of 3 we can make:

      157894736842105263 = n
      315789473684210526 = n × 2
      631578947368421052 = n × 4

      These numbers are the repetends of fractions of the form k / 19.

      Like

    • Frits's avatar

      Frits 8:51 am on 31 January 2026 Permalink | Reply 

      '''
  n = xbcd where x is a k-digit number and b,c and d are digits 
2.n = dxbc
4.n = cdxb
8.n = bcdx with x, b and c all even

8.n = bcdx also has k+3 digits so b = 8 and first digit of x = 1
   --> c = 4 (not 9) and last digit of x must be 6
       d = 2 (as 2.n must be 2.....)
       
n = 1....6842      

  n = xbcd = 1000.x + bcd  (x has k digits)
8.n = bcdx = 10^k . bcd + x

8000.x + 8.bcd = 10^k . bcd + x
7999.x = 19.421.x = (10^k - 8).842
19.x = 2.(10^k - 8)
'''
      
for k in range(2, 18):  
  x, r = divmod(2 * (10**k - 8), 19)
  if r: continue
  n = str(x) + "842"
  # no digit occurs more than twice
  if all(n.count(str(i)) <= 2 for i in range(10)): 
    print("answer method 1:", n)
  
# in order for 2 * (10**k - 8) % 19 = 0 we must have
# 2 * 10**k % 19 = 16 
# 2 * 10**k = 19.10**(k - 1) + 10**(k - 1)
# so 10**(k - 1) % 19 = 16

# 10**(k - 1) % 19 follows a logical pattern
k = 2
a = 10**(k - 1) % 19
while a != 16: 
  a = (a + 19 * (a % 2)) // 2
  k += 1

n = str(2 * (10**k - 8) // 19) + "842"
# no digit occurs more than twice
if all(n.count(str(i)) <= 2 for i in range(10)): 
  print("answer method 2:", n)

      Like

  • Unknown's avatar

    Jim Randell 8:11 am on 25 January 2026 Permalink | Reply
    Tags:   

    Teaser 3305: Things with scales and things to scale with 

    From The Sunday Times, 25th January 2026 [link] [link]

    Rummaging through a cupboard I came across an old Snakes and Ladders set. The ladders (label them A to E) went from square 4 to 26, 18 to 28, 19 to 73, 22 to 60, and 41 to 98. The snakes (label them W to Z) went from square 53 to 24, 58 to 5, 72 to 33, and 90 to 66. For old times’ sake I couldn’t resist playing a game on my own. Starting from square 1, rolling the die and moving appropriately, travelling travelling up ladders and down snakes when I chanced to land on their starting squares, I happened to finish exactly on square 100. The total of all the numbers I had thrown on the die was 51.

    What was the order of all my up and down jumps? (e.g. A, Y, D).

    [teaser3305]

     
    • Jim Randell's avatar

      Jim Randell 8:56 am on 25 January 2026 Permalink | Reply

      Here is a program that attempts to constructively play games that finish exactly on square 100 and hve a throw total of 51.

      We assume that once a possible game is found, all other possible games have the same sequence of jumps. (I’m thinking about better ways to show the answer to the puzzle is unique).

      It runs in 77ms. (Internal runtime is 1.1ms).

      from enigma import (tuples, seq2str, printf)

# snakes and ladders
jumps = {
  # ladders (lower -> higher)
  (4, 26): 'A', (18, 28): 'B', (19, 73): 'C', (22, 60): 'D', (41, 98): 'E',
  # snakes (higher -> lower)
  (53, 24): 'W', (58, 5): 'X', (72, 33): 'Y', (90, 66): 'Z',
}

# linked squares
link = dict(jumps.keys())

# possible throws
throws = [6, 5, 4, 3, 2, 1]

# play the game
# p = current position
# t = target position
# b = throw budget (must not be exceeded)
# ds = dice throws
# ps = positions
def play(p, t, b, ds=[], ps=[]):
  ps = ps + [p]
  # are we done
  if p == t or p >= 100 or b <= 0:
    yield (ds, ps, b)
  elif b > 0:
    # make the next roll
    for d in throws:
      if not (d > b):
        # move the player
        p1 = p + d
        # perform any jump
        p2 = link.get(p1)
        # continue play
        if p2 is None:
          yield from play(p1, t, b - d, ds + [d], ps)
        else:
          yield from play(p2, t, b - d, ds + [d], ps + [p1])

# find a possible game
for (ds, ps, r) in play(1, 100, 51):
  if r == 0 and ps[-1] == 100:
    printf("throws = {ds} (sum = {t}) -> positions = {ps}", t=sum(ds))
    # find the jumps involved
    js = tuple(filter(None, (jumps.get(k) for k in tuples(ps, 2))))
    printf("jumps = {js}", js=seq2str(js, enc="[]"))
    break

      Solution: The jumps made are: C, Z, Y, E.

      So the game proceeds as follows:

      dice throws of 18: move 1 → 19 (avoiding 4, 18)
      take ladder C from 19 → 73
      dice throws of 17: move 73 → 90
      take snake Z from 90 → 66
      dice throws of 6: move 66 → 72
      take snake Y from 72 → 33
      dice throws of 8: move 33 → 41
      take ladder E from 41 → 98
      dice throws of 2: move 98 → 100
      total dice throws = 18 + 17 + 6 + 8 + 2 = 51

      Like

      • Jim Randell's avatar

        Jim Randell 1:28 pm on 25 January 2026 Permalink | Reply

        Here is a better solution. It is shorter, faster, can be used to experiment with different throw budgets, and it finds all solutions (thereby showing the answer to the original puzzle is unique).

        We don’t care about the actual dice throws, a game is made up of a sequence of contiguous blocks (that are moved with throws of the dice) with jumps (i.e. a snake or a ladder) between the blocks. The blocks themselves can usually be made by lots of dice throws (and the starts of the jumps are sufficiently fragmented that we can always avoid any unwanted jumps).

        This Python program makes a collection of possible contiguous segments, and then joins a sequence of them together with jumps inbetween to make a game that starts at 1, ends at 100, and requires a total of 51 to be scored on the dice throws during the segments.

        It runs in 71ms. (Internal runtime is 88µs).

        from enigma import (defaultdict, tuples, seq2str, printf)

# snakes and ladders
jumps = {
  # ladders (lower -> higher)
  (4, 26): 'A', (18, 28): 'B', (19, 73): 'C', (22, 60): 'D', (41, 98): 'E',
  # snakes (higher -> lower)
  (53, 24): 'W', (58, 5): 'X', (72, 33): 'Y', (90, 66): 'Z',
}

# linked squares
link = dict(jumps.keys())

# possible contiguous segments; segs: start -> ends
segs = defaultdict(list)
seg = lambda x, y: segs[x].append(y)
seg(1, 100)  # no jumps
for (a, b) in jumps.keys():
  # add in 1 to the start of a jump
  seg(1, a)
  # add in end of a jump to 100
  seg(b, 100)
  # add in end of a jump to start of a jump
  for (x, y) in jumps.keys():
    if b < x: seg(b, x)

# find contiguous segments from <s> to <t> with throw budget <b>
def solve(s, t, b, ss=[]):
  # are we done?
  if s == t and b == 0:
    yield ss
  elif s < 100 and b > 0:
    # add in the next segment
    for x in segs[s]:
      d = x - s
      if not (d > b):
        yield from solve(link.get(x, x), t, b - d, ss + [(s, x)])

# find games from 1 to 100 with a throw budget of B
B = 51
for ss in solve(1, 100, B):
  # find jumps used
  js = tuple(jumps[(b, x)] for ((a, b), (x, y)) in tuples(ss, 2))
  printf("{B}: {ss} -> {js}", js=seq2str(js, enc="[]"))

        Liked by 1 person

  • Unknown's avatar

    Jim Randell 9:48 am on 23 January 2026 Permalink | Reply
    Tags:   

    Teaser 2462: [What NOW?] 

    From The Sunday Times, 29th November 2009 [link]

    Puzzlers are often confused about whether the number 1 is prime or square, so perhaps this Teaser will clarify the issue!

    • ONE is not prime, but it is an odd perfect square;
    • TWO is divisible by 2 but not 4;
    • The number of odd primes less than SIX is TWO.

    In those statements digits have consistently been replaced by capital letters, with different letters being used for different digits.

    What is the number NOW?

    This puzzle was originally published with no title.

    [teaser2462]

     
    • Jim Randell's avatar

      Jim Randell 9:49 am on 23 January 2026 Permalink | Reply

      Here is a solution using the [[ SubstitutedExpression ]] solver from the enigma.py library.

      The following run-file executes in 98ms. (Internal runtime of the generated code is 14.4ms).

      #! python3 -m enigma -rr

SubstitutedExpression

# no leading zeros (ONE, TWO, SIX, NOW)
--invalid="0,OTSN"

# collect primes < 1000
--code="primes.expand(999)"

# ONE is not prime, but it is an odd perfect square
"ONE not in primes"
"E % 2 = 1"
"is_square(ONE)"

# TWO is divisible by 2, but not 4
"O % 2 = 0"
"WO % 4 != 0"

# the number of odd primes less than SIX is TWO
"icount(primes.between(3, SIX - 1)) = TWO"

--template="(ONE) (TWO) (SIX)"
--answer="NOW"

      Solution: NOW = 820.

      There are two possible assignments of digits to letters:

      ONE = 289, TWO = 102, SIX = 564
      ONE = 289, TWO = 102, SIX = 567

      The next prime after 563 is 569, so we count 102 odd primes if X is 4 or 7.

      Like

      • Frits's avatar

        Frits 9:09 am on 24 January 2026 Permalink | Reply

        @Jim, your code allows N to be zero. Is this what you wanted?

        Like

    • ruudvanderham's avatar

      ruudvanderham 11:24 am on 23 January 2026 Permalink | Reply 

      import peek
import istr

for o, n, e, t, w, s, i, x in istr.permutations(range(10), 8):
    if o * t * s != 0:
        if not (one := istr("=one")).is_prime() and one.is_odd() and one.is_square():
            if (two := istr("=two")).is_divisible_by(2) and not two.is_divisible_by(4):
                if len(istr.primes(3, (six := istr("=six")))) == two:
                    peek(one, two, six)

      Like

  • Unknown's avatar

    Jim Randell 9:16 am on 20 January 2026 Permalink | Reply
    Tags:   

    Teaser 2363: [A round number] 

    From The Sunday Times, 6th January 2008 [link]

    We are used to thinking of 1,000 as a “round number”, but this is very much based on 10. One definition of a round number is that it should have lots of small factors, and certainly that it should be divisible by every number up to and including sixteen. There is one such round number that, when written in a certain base (of less than 10,000), looks like 1111. What is more, its base can be found very neatly.

    What is that base?

    This puzzle was originally published with no title.

    [teaser2363]

     
    • Jim Randell's avatar

      Jim Randell 9:17 am on 20 January 2026 Permalink | Reply

      Programatically, it is straightforward to just try all possible bases.

      This Python program runs in 75ms. (Internal runtime is 4.2ms).

      from enigma import (irange, mlcm, call, nconcat, printf)

# M = LCM(2..16)
M = call(mlcm, irange(2, 16))

# consider possible bases
for b in irange(2, 9999):
  # n = 1111 [base b]
  n = nconcat([1, 1, 1, 1], base=b)
  # check for divisibility by 2..16
  if n % M == 0:
    # output solution
    printf("base={b} n={n}")

      But here is a more sophisticated approach:

      Another way to write 1111 [base b] is: b³ + b² + b + 1, which is an integer valued polynomial function.

      And this means we can use the [[ poly_roots_mod() ]] solver from the enigma.py library (described here: [ link ]) to solve the puzzle.

      The following Python program has an internal runtime of 572µs.

      from enigma import (Polynomial, rev, irange, mlcm, call, poly_roots_mod, printf)

# calculate 1111 [base b]
p = Polynomial(rev([1, 1, 1, 1]))

# M = LCM(2..16)
M = call(mlcm, irange(2, 16))

# find roots of p(b) = 0 [mod M] that give a valid base
roots = poly_roots_mod(p, 0)
for b in roots(M):
  if 1 < b < 10000:
    # calculate the corresponding n value
    n = p(b)
    # output solution
    printf("base={b} n={n}")

      Solution: The base is 5543.

      And the round number is: 1111 [base 5543] = 170338568400 [decimal] = 720720 × 236345.

      Like

    • Ruud's avatar

      Ruud 10:52 am on 20 January 2026 Permalink | Reply 

      import istr

print(*(base for base in istr.range(2, 10000) if all(sum(base**i for i in range(4)).is_divisible_by(i) for i in range(2, 17))))

      Like

    • Frits's avatar

      Frits 8:33 am on 21 January 2026 Permalink | Reply 

      # 1111 [base b] is: b^3 + b^2 + b + 1 or (b + 1)(b^2 + 1)
# I am not good at modular arithmetic but b^2 + 1 is never a multiple of
# numbers 4.k + 3. So b + 1 must be divisible by 3, 7 and 11 and thus by 231.

for b1 in range(231, 10001, 231):
  # calculate n = 1111 [base b]
  b = b1 - 1
  n = b1 * (b**2 + 1)
  # n must be divisible by numbers 2-16
  if any(n % i for i in [9, 11, 13, 14, 15, 16]): continue
  print("answer:", b)

      Like

      • Jim Randell's avatar

        Jim Randell 9:11 am on 21 January 2026 Permalink | Reply

        @Frits: A neat bit of analysis. That is probably the kind of approach the setter had in mind.

        In fact as (b² + 1) is not divisible by 3, then it is also not divisible by 9, so we can proceed in steps of: 9 × 7 × 11 = 693, which only needs 14 iterations.

        This Python program has an internal runtime of 34µs:

        from enigma import (irange, printf)

S = 9 * 7 * 11
R = 720720 // S
for i in irange(S, 10000, step=S):
  b = i - 1
  n = i * (b*b + 1)
  if n % R == 0:
    printf("base={b} n={n}")

        Like

        • Jim Randell's avatar

          Jim Randell 9:40 am on 21 January 2026 Permalink | Reply

          In fact although (b² + 1) may be divisible by 2, it cannot be divisible by 4, so we can move in steps of:

          S = 8 × 9 × 7 × 11 = 5544

          And, as the base is limited to be less than 10000, there is only one value to check (or if we accept that there is a solution to the puzzle, then there are no values to check).

          Like

          • Frits's avatar

            Frits 1:10 pm on 21 January 2026 Permalink | Reply

            @Jim, good work.

            if b is even then b=2k and b^2 + 1 = 4k^2 + 1
            if b is odd then b = 2k+1 and b^2 + 1 = 4k^2 + 4k + 2

            so indeed b^2 + 1 cannot be divisible by 4.

            Like

  • Unknown's avatar

    Jim Randell 6:38 am on 18 January 2026 Permalink | Reply
    Tags:   

    Teaser 3304: Roaming bull 

    From The Sunday Times, 18th January 2026 [link] [link]

    Farmer Quentin keeps his prize bull in a circular field of diameter 50m. To improve the layout, he decides to switch to a square field with side length equal to the old diameter. He builds the new fencing around the old, with fence posts every metre starting at the corner. After completing the new fencing, he tethers the bull outside the fence to one of these fence posts with a rope of integer metre length, to allow the removal of the inner circular fence. As a result, the bull’s temporary roaming area is now four times what it was inside the circle.

    What is the length of the rope?

    [teaser3304]

     
    • Jim Randell's avatar

      Jim Randell 7:14 am on 18 January 2026 Permalink | Reply

      If the bull were tethered to a single post with a 50 m rope in an otherwise unobstructed flat plain, then it would be able to roam a region that is 4 times the area of the original (25 m radius) circular field. So the rope must be at least this long.

      When tethered to a post that makes up the boundary of the square field, then bull can roam over a collection of quadrants of a circle are centred on the tether point, and the corners of the square field, as the rope wraps around the boundary of the field.

      I considered rope lengths allowing roaming areas of between 2 and 6 non-overlapping quadrants (of various sizes) around the outside of the field. (Which assumes the bull has no other obstructions). If the rope is longer than 100 m then the regions will overlap, as the bull can reach the opposite point on the perimeter of the square from either direction.

      For each possible tethering point a corresponding equation is generated and then solved to find possible integer rope lengths.

      The following Python code runs in 79ms. (Internal runtime is 2.1ms).

      from enigma import (Rational, Polynomial as Poly, irange, sq, divc, sprintf as f, printf)

Q = Rational()
q = Q(1, 4)

d = 50  # size of the square
A = sq(d)  # target area (= 4 * r^2 where r = d/2)

# find roots of polynomial <p> = <v> between <lo> and <hi>
def roots(p, v, lo, hi, s=''):
  xs = p.roots(v, domain='Z', include='+', F=Q)  # integer roots
  for x in xs:
    if lo <= x <= hi:
      printf("{s}x={x}")

# suppose the bull is tethered at post a = 0 (corner) to 25 (middle)
# with a rope of length x
for a in irange(0, divc(d, 2)):

  # 2 quadrants: 0 <= x <= a
  p = 2*q * sq(Poly([0, 1]))  # x^2
  roots(p, A, 0, a, f("[2 quads] a={a} -> "))

  # 3 quadrants: a <= x <= d - a
  p += q * sq(Poly([-a, 1]))  # (x - a)^2
  roots(p, A, a, d - a, f("[3 quads] a={a} -> "))

  # 4 quadrants: d - a <= x <= d + a
  p += q * sq(Poly([a - d, 1]))  # (x + (a - d))^2
  roots(p, A, d - a, d + a, f("[4 quads] a={a} -> "))

  # 5 quadrants: d + a <= x <= 2d - a
  p += q * sq(Poly([-(a + d), 1]))  # (x - (a + d))^2
  roots(p, A, d + a, 2*d - a, f("[5 quads] a={a} -> "))

  # 6 quadrants (no overlap): 2d - a <= x <= 2d
  p += q * sq(Poly([a - 2*d, 1]))  # (x + (a - 2d))^2
  roots(p, A, 2*d - a, 2*d, f("[6 quads] a={a} -> "))

      Solution: The length of the rope is 58 m.

      And the tether is at post 2 m from a corner.

      The bull can roam in two quadrants with radius 58 m, one quadrant with radius 56 m, one quadrant with radius 10 m, and one quadrant with radius 6 m.

      Here is a diagram of the situation – the light green area (outside the square field) is 4 times the size of the dark green area (inside the square field):

      If Farmer Quentin only measures the rope to approximately a whole number of metres, then there are two more solutions where the rope is within 4 cm of a whole number of metres:

      rope = 58.99 m; tether = 6 m from a corner
      rope = 60.03 m; tether = 12 m from a corner

      We can see these non-integer solutions by setting [[ domain='F', F='float' ]] at line 11 in the program above.

      Like

      • Jim Randell's avatar

        Jim Randell 11:00 am on 18 January 2026 Permalink | Reply

        Alternatively, just considering all possible rope lengths and tether points (although this doesn’t let you find non-integer rope lengths).

        This Python program runs in 392µs.

        from enigma import (irange, sq, divc, printf)

d = 50  # size of the square
A = 4 * sq(d)  # target area (in quarter units)

# consider possible tether points
for a in irange(0, divc(d, 2)):

  # consider increasing rope length
  for x in irange(d, 2*d):

    # calculate total area of possible quadrants
    rs = [x, x, x - a, x + a - d, x - a - d, x + a - 2*d]
    v = sum(sq(r) for r in rs if r > 0)

    # output solution
    if v == A: printf("a={a} x={x} [quads={q}]", q=sum(r > 0 for r in rs))
    if v >= A: break

        Like

    • Frits's avatar

      Frits 1:43 pm on 18 January 2026 Permalink | Reply

      3 areas (or 4 quadrants) will definitely be roamed as I don’t let the rope length start from 1.
      A 6th quadrant is not possible for diameter 50 as I have calculated a basic upper limit for the rope length.

      d = 50 # diameter in meters

# old roaming area = 25^2.pi so new roaming area must be 50^2.pi
# southern area S = r^2 / 2 < 2500, r^2 < 5000
mx_rope = int((2 * d**2)**.5)

# the rope length must be larger than the diameter
for r in range(d + 1, mx_rope + 1):
  # tether the bull to the southern fence (on the west side)
  
  for y in range(26):   # distance from SW corner
    # quadruple areas divided by pi
    S4 = 2 * r**2        # 4 x area below the southern fence
    NW4 = (r - y)**2 
    N4 = (r - d - y)**2 if r - d - y > 0 else 0
    E4 = (r - d + y)**2 
    
    # all areas should sum to 100^2
    if S4 + NW4 + N4 + E4 == 4 * d**2:
      print("answer:", r, "meters")

      Like

  • Unknown's avatar

    Jim Randell 8:11 am on 16 January 2026 Permalink | Reply
    Tags: ,   

    Teaser 2312: Backward step 

    From The Sunday Times, 14th January 2007 [link]

    Seb had just been given his four-digit PIN, and he was afraid that he would not be able to remember it. So he broke all the rules and wrote it in his pocket diary. But to make it “more secure” he did not actually write the PIN; instead, he wrote the digits in reverse order. He was surprised to discover that the new number was a multiple of the correct PIN.

    What is his PIN?

    Note that without additional requirements there are many solutions to this puzzle. The intended solution seems to come from the requirement that all the digits of the PIN are non-zero.

    [teaser2312]

     
    • Jim Randell's avatar

      Jim Randell 8:12 am on 16 January 2026 Permalink | Reply

      Here is a solution using the [[ SubstitutedExpression ]] solver from the enigma.py library.

      It runs in 82ms. (Internal runtime of the generated code is 1.7ms).

      #! python3 -m enigma -rr

SubstitutedExpression

# suppose the real PIN is ABCD
"ediv(DCBA, ABCD) > 1"
"D >= A" # [optional]

--distinct=""
--invalid=""
--digits="1-9"
--answer="ABCD"

      Solution: The PIN is: 2178.

      And so the reversed PIN is: 8712.

      And: 8712 = 4 × 2178.

      (The multiple solutions to the puzzle as published can be seen by removing the digit restriction on line 11).

      Without the restriction to non-zero digits there are 131 solutions, for example (reversed-PIN, PIN):

      1000 & 0001
      1010 & 0101
      …
      9801 & 1089
      …
      9990 & 0999

      (Although not all these would be surprising).

      But only one of these solutions remains if the digits of the PIN must all be non-zero.

      The published solution was: “1089 or 2178”.

      Like

    • Ruud's avatar

      Ruud 10:33 am on 16 January 2026 Permalink | Reply 

      import istr
print(*(pin for digits in istr.product(range(1, 10), repeat=4) if (pin:=istr.join(digits))[::-1].divided_by(pin) not in (1, None)))

      Like

    • GeoffR's avatar

      GeoffR 7:43 pm on 16 January 2026 Permalink | Reply 

      from enigma import nsplit, is_pairwise_distinct

for pin in range(1234, 9876):
    a, b, c, d = nsplit(pin)
    if 0 in (a, b, c, d): continue
    if is_pairwise_distinct(a, b, c, d):
        rev_pin = 1000*d + 100*c + 10*b + a
        q, r = divmod(rev_pin, pin)
        if r == 0 and q > 1:
           print(f"Pin = {pin}.")

      Like

  • Unknown's avatar

    Jim Randell 3:34 pm on 14 January 2026 Permalink | Reply
    Tags:   

    Brain teaser 983: Superchamp 

    From The Sunday Times, 24th May 1981 [link]

    The five champion athletes competed against one another in their five special events to decide who was the supreme champion (the one with the highest number of points). Three points were awarded for the winner in each event. two points for second place, and one for third.

    The Superchamp won more events than anyone else. All scored a different number of points, and all scored in a different number of events.

    Ben and Colin together scored twice as many points as Fred. Eddie was the only one to win his own special event, and in it the Superchamp came second. In another event Eddie gained one point less than the hurdler. The hurdler only came third in the hurdles, which was won by Denis, who was last in the long jump but scored in the sprint. Fred won the long jump and was second in the walk, but came nowhere in the hurdles.

    Ben scored in the long jump and Colin scored in the sprint. The thousand-metre man was third in two events. The long jumper scored more points than the sprinter.

    Who was the [long] jumper, and what was his score?

    This puzzle is included in the book The Sunday Times Book of Brainteasers (1994).

    [teaser983]

     
    • Jim Randell's avatar

      Jim Randell 3:35 pm on 14 January 2026 Permalink | Reply

      There are a lot of conditions to this puzzle, so I used the minizinc.py library to allow me to construct a declarative model for it in MiniZinc, which is then solved, and the Python program used to format the solutions.

      It runs in 545ms.

      from enigma import (require, printf)
from minizinc import MiniZinc

require("sys.version", "3.6") # needed to support [ use_embed=1 ]

# indices for names and events (1..5)
(B, C, D, E, F) = (1, 2, 3, 4, 5)
(hurdles, long_jump, sprint, m1000, walk) = (1, 2, 3, 4, 5)

# construct the MiniZinc model
m = MiniZinc([
  'include "globals.mzn"',

  # scores for the competitors in each event: pts[event, competitor] = score
  "array [1..5, 1..5] of var 0..3: pts",

  # scores in each event are: 3, 2, 1, 0, 0
  "constraint forall (i in 1..5) (sum (j in 1..5) (pts[i, j] = 3) = 1)", # 1st = 3pts
  "constraint forall (i in 1..5) (sum (j in 1..5) (pts[i, j] = 2) = 1)", # 2nd = 2pts
  "constraint forall (i in 1..5) (sum (j in 1..5) (pts[i, j] = 1) = 1)", # 3rd = 1pts
  "constraint forall (i in 1..5) (sum (j in 1..5) (pts[i, j] = 0) = 2)", # unplaced = 0pts

  # functions:
  "function var int: points(var int: j) = sum (i in 1..5) (pts[i, j])",
  "function var int: wins(var int: j) = sum (i in 1..5) (pts[i, j] = 3)",
  "function var int: scored(var int: j) = sum (i in 1..5) (pts[i, j] > 0)",

  # "the superchamp scored more points than anyone else"
  "var 1..5: S",
  "constraint forall (j in 1..5 where j != S) (points(j) < points(S))",

  # "the superchamp won more events than anyone else"
  "constraint forall (j in 1..5 where j != S) (wins(j) < wins(S))",

  # each competitor scored a different number of points
  "constraint all_different (j in 1..5) (points(j))",

  # each competitor scored in a different number of events
  "constraint all_different (j in 1..5) (scored(j))",

  # "B and C together scored twice as many points as F"
  "constraint points({B}) + points({C}) = 2 * points({F})",

  # "E was the only one to win his own event"
  # "and in that event the superchamp came second"
  "array [1..5] of var 1..5: spe",
  "constraint all_different(spe)",
  "constraint forall (j in 1..5) ((j == {E}) <-> (pts[spe[j], j] = 3))",
  "constraint pts[spe[{E}], S] = 2",

  # "in another event E gained 1 point less than the hurdler"
  "var 1..5: hurdler",
  "constraint spe[hurdler] = {hurdles}",
  "constraint exists (i in 1..5) (pts[i, {E}] + 1 = pts[i, hurdler])",

  # "the hurdler only came third in hurdles"
  "constraint pts[{hurdles}, hurdler] = 1",

  # "hurdles was won by D"
  "constraint pts[{hurdles}, {D}] = 3",

  # "D came last in long jump"
  "constraint pts[{long_jump}, {D}] = 0",

  # "D scored in the sprint"
  "constraint pts[{sprint}, {D}] > 0",

  # "F won the long jump"
  "constraint pts[{long_jump}, {F}] = 3",

  # "F was 2nd in the walk"
  "constraint pts[{walk}, {F}] = 2",

  # "F did not score in hurdles"
  "constraint pts[{hurdles}, {F}] = 0",

  # "B scored in the long jump"
  "constraint pts[{long_jump}, {B}] > 0",

  # "C scored in the sprint"
  "constraint pts[{sprint}, {C}] > 0",

  # "the 1000m man was 3rd in 2 events"
  "var 1..5: m1000er",
  "constraint spe[m1000er] = {m1000}",
  "constraint sum (i in 1..5) (pts[i, m1000er] = 1) = 2",

  # "the long jumper scored more points than the sprinter"
  "var 1..5: long_jumper",
  "constraint spe[long_jumper] = {long_jump}",
  "var 1..5: sprinter",
  "constraint spe[sprinter] = {sprint}",
  "constraint points(long_jumper) > points(sprinter)",

], use_embed=1)


# map indices back to names / events (for output)
name = {1: "Ben", 2: "Colin", 3: "Denis", 4: "Eddie", 5: "Fred"}
event = {1: "hurdles", 2: "long jump", 3: "sprint", 4: "1000m", 5: "walk"}

# solve the model and find:
# pts = results in each event
# spe = special event for each competitor
for (pts, spe) in m.solve(solver="minizinc -a", result="pts spe"):
  # collect points
  t = dict((k, 0) for k in name.keys())

  # output the results for each event
  for (i, rs) in enumerate(pts, start=1):
    (p1, p2, p3) = (rs.index(p) + 1 for p in (3, 2, 1))
    printf("{e}: 1st = {p1}, 2nd = {p2}, 3rd = {p3}", e=event[i], p1=name[p1], p2=name[p2], p3=name[p3])
    # collect points
    t[p1] += 3
    t[p2] += 2
    t[p3] += 1
  printf()

  # determine superchamp
  S = max(t.keys(), key=t.get)
  assert all(k == S or v < t[S] for (k, v) in t.items())

  # output the results for each competitor
  for j in sorted(event.keys()):
    printf("{j} [{e}]: {t} points{S}", j=name[j], e=event[spe[j - 1]], t=t[j], S=(' [SUPERCHAMP]' if j == S else ''))
  printf("--")
  printf()

      Solution: Denis is the long jumper. He scored 5 points.

      The program finds 2 possible scenarios:

      hurdles: 1st = Denis, 2nd = Eddie, 3rd = Ben
      long jump: 1st = Fred, 2nd = Ben, 3rd = Eddie
      sprint: 1st = Ben, 2nd = Denis, 3rd = Colin
      1000m: 1st = Eddie, 2nd = Ben, 3rd = Fred
      walk: 1st = Ben, 2nd = Fred, 3rd = Eddie

      Ben [hurdles]: 11 points [SUPERCHAMP]
      Colin [sprint]: 1 points
      Denis [long jump]: 5 points
      Eddie [1000m]: 7 points
      Fred [walk]: 6 points

      hurdles: 1st = Denis, 2nd = Eddie, 3rd = Colin
      long jump: 1st = Fred, 2nd = Ben, 3rd = Colin
      sprint: 1st = Colin, 2nd = Denis, 3rd = Eddie
      1000m: 1st = Eddie, 2nd = Colin, 3rd = Fred
      walk: 1st = Colin, 2nd = Fred, 3rd = Eddie

      Ben [sprint]: 2 points
      Colin [hurdles]: 10 points [SUPERCHAMP]
      Denis [long jump]: 5 points
      Eddie [1000m]: 7 points
      Fred [walk]: 6 points

      The solution published in the book (which runs to 3 pages) only gives the first of these solutions.

      I think this is due to the interpretation of “In another event Eddie gained one point less than the hurdler”. The book solution seems to infer (hurdler = 3pts, Eddie = 2pts) or (hurdler = 2pts, Eddie = 1pt), and concludes it is not possible for C to be the champ.

      And in the first scenario, in the long jump, we have: the hurdler (Ben) scores 2pts (2nd place), and Eddie scores 1pt (3rd place).

      However, I think the wording also allows (hurdler = 1pt, Eddie = 0pts), and in the second scenario, in the long jump, we have: the hurdler (Colin) scores 1pt (3rd place), and Eddie scores 0pts (unplaced).

      Like

    • Frits's avatar

      Frits 3:06 pm on 17 January 2026 Permalink | Reply 

      from itertools import permutations

'''        3   2   1       
hurdles    A   F   K        
long J     B   G   L        
sprint     C   H   M              super=Superchamp
1000m      D   I   N        
Walk       E   J   O        
'''

sports = [-1, -1, -1, -1, -1]
(ben, colin, denis, eddie, fred) = range(5)

dgts, sols = set(range(5)), set()

A = denis # Denis won hurdles
B = fred  # Fred won the long jump
J = fred  # Fred was 2nd in the walk

# Fred came nowhere in the hurdles
for F, K in permutations(dgts - {A, fred}, 2):    # hurdles
  # hurdler (K) came 3rd in the hurdles, Eddie is no hurdler
  if eddie == K: continue
  for G, L in permutations(dgts - {B, denis}, 2): # Denis last in long jump
    if 0 not in {G, L} : continue                 # Ben scored in long jump
    for C, H, M in permutations(dgts, 3):         # sprint
      # Colin and Denis scored in the sprint
      if not {C, H, M}.issuperset({colin, denis}): continue 
      for D, I, N in permutations(dgts, 3):       # 1000m
        for E, O in permutations(dgts - {J}, 2):  # walk
          gold   = [A, B, C, D, E]
          silver = [F, G, H, I, J]
          bronze = [K, L, M, N, O]
          # Eddie was the only one to win his own special event
          if eddie not in gold: continue
          scores = [A, F, K, B, G, L, C, H, M, D, I, N, E, J, O]
          scored = [scores.count(i) for i in range(5)]
          # all scored in a different number of events.
          if len(set(scored)) != 5: continue
          
          pts = [sum([3 - (i % 3) for i, s in enumerate(scores) if s == j]) 
                 for j in range(5)]
          # all scored a different number of points
          if len(set(pts)) != 5: continue
          # Ben and Colin together scored twice as many points as Fred
          if pts[ben] + pts[colin] != 2 * pts[fred]: continue
          # the Superchamp won more events than anyone else ( >= 2)
          sups = sorted((f, i) for i in range(5) if (f := gold.count(i)) > 1)
          if not sups or (len(sups) == 2 and len(set(gold)) == 3): continue
          
          super = sups[-1][1]
          # once the Superchamp came 2nd after Eddie
          if super == eddie or super not in silver: continue
          # the Superchamp had the highest number of points
          if pts[super] != max(pts): continue
          
          sports[0] = K  # hurdler only came third in the hurdles
                
          # check sports where Eddie and Superchamp ended 1 and 2
          for e, (g, s) in enumerate(zip(gold, silver)):
            if (g, s) != (eddie, super): continue
            
            if e == 0: continue # Eddie is not the hurdler
            # in another event Eddie gained one point less than the hurdler
            for i, gsb in enumerate(zip(gold, silver, bronze)):
              if ((K, eddie) in zip(gsb, gsb[1:]) or 
                  (eddie not in gsb and gsb[-1] == K)): break
            else: # no break
              break    
            
            sports_ = sports.copy()
            sports_[e] = eddie  # Eddie performs sport <e>
            # assign people to remaining sports
            for p in permutations(dgts - {eddie, K}):
              sports__ = sports_.copy()
              j = 0
              for i in range(5):
                if sports_[i] != -1: continue
                sports__[i] = p[j]
                j += 1
              
              # Eddie was the only one to win his own special event
              for i, s in enumerate(sports__):
                if gold[i] == s and s != eddie: break
              else: # no break  
                # the long jumper scored more points than the sprinter
                if not (pts[sports__[1]] > pts[sports__[2]]): continue
                # the thousand-metre man was third in two events
                if bronze.count(sports__[3]) != 2: continue
                # store solution
                sols.add((sports__[1], pts[sports__[1]]))

names = "Ben Colin Denis Eddie Fred".split()                
print("answer:", ' or '.join(names[a] + ' with score ' + str(s) 
                              for a, s in sols))

      Like

  • Unknown's avatar

    Jim Randell 6:25 am on 11 January 2026 Permalink | Reply
    Tags:   

    Teaser 3303: Knights and knaves 

    From The Sunday Times, 11th January 2026 [link] [link]

    On my travels, I came across a group of some knights (always truthful) and some knaves (always lying). At first, I couldn’t tell which were knights and which were knaves. However, six individuals then gave the following statements about their group:

    “The number of knights is prime.”
    “The number of knights is odd.”
    “The number of knaves is square.”
    “The number of knaves is even.”
    “The total of knights and knaves is an odd number.”
    “There are more knaves than knights.”

    These statements enabled me to deduce the composition of the group.

    How many (a) knights and (b) knaves were in the group?

    [teaser3303]

     
    • Jim Randell's avatar

      Jim Randell 6:47 am on 11 January 2026 Permalink | Reply

      The setter has an additional piece of information – i.e. they can observe the total number of people in the group. If it is possible to break this total down in into knights and knaves in only one way that is consistent with the statements made, then the composition of the group can be determined with certainty.

      The following Python program runs in 71ms. (Internal runtime is 158µs).

      from enigma import (irange, inf, decompose, filter2, is_prime, is_square_p, printf)

# consider increasing group sizes (at least 6)
for t in irange(6, inf):
  # collect candidate solutions for this total size
  sols = list()
  # decompose into (a) knights and (b) knaves
  for (a, b) in decompose(t, 2, min_v=1, increasing=0, sep=0):

    # evaluate the statements
    ss = filter2(bool, [
      # 1. "a is prime"
      is_prime(a),
      # 2. "a is odd"
      (a % 2 == 1),
      # 3. "b is square"
      is_square_p(b),
      # 4. "b is even"
      (b % 2 == 0),
      # 5. "a + b is odd"
      (t % 2 == 1),
      # 6. "b > a"
      (b > a),
    ])

    # check minimum group sizes
    if len(ss.true) > a or len(ss.false) > b: continue
    # record this candidate
    sols.append((a, b))

  printf("[{t} -> {sols}]")
  # check for unique solutions
  if len(sols) == 1:
    (a, b) = sols[0]
    printf("a = {a}; b = {b}")
    break

      Solution: (a) There are 5 knights; (b) There are are 2 knaves.

      The total size of the group is 7, and the statements made are:

      1. True (“the number of knights is prime” = 5)
      2. True (“the number of knights is odd” = 5)
      3. False (“the number of knaves is square” = 2)
      4. True (“the number of knaves is even” = 2)
      5. True (“the total number of knights and knaves is odd” = 7)
      6. False (“there are more knaves than knights” 5 > 2)

      There are 4 true statements, and 2 false statements, so both knaves made a false statement, and 4 of the 5 knights made a true statement.

      And this is the only way a group of 7 can be broken down consistent with the statements.

      If we allow the program to continue to larger group sizes, we see that the number of candidate breakdowns grows, so it would not be possible to determine the composition of a larger group.

      Like

  • Unknown's avatar

    Jim Randell 9:58 pm on 9 January 2026 Permalink | Reply
    Tags:   

    Teaser 2360: [Flower arranging] 

    From The Sunday Times, 16th December 2007 [link]

    In the following statements and instruction, digits have been consistently replaced by letters, with different letters for different digits.

    F + L + O + W + E + R raised to the power P is equal to the six-digit number FLOWER.

    FACTOR is divisible by A, R and T.

    What is the value of FLOWER ART?

    This puzzle was originally published with no title.

    [teaser2360]

     
    • Jim Randell's avatar

      Jim Randell 10:03 pm on 9 January 2026 Permalink | Reply

      Here is a solution using the [[ SubstitutedExpression ]] solver from the enigma.py library.

      It runs in 226ms. (Internal runtime of the generated code is 126ms).

      #! python3 -m enigma -rr

SubstitutedExpression

"pow(F + L + O + W + E + R, P) = FLOWER"

"FACTOR % A = 0"
"FACTOR % R = 0"
"FACTOR % T = 0"

--answer="(FLOWER, ART)"

      Solution: FLOWER ART = 390625 751.

      And:

      FLOWER = 390625 = 25^4 = (3 + 9 + 0 + 6 + 2 + 5) ^ 4.
      FACTOR = 378105 is divisible by 7, 5, 1.

      Like

    • Ruud's avatar

      Ruud 6:11 am on 10 January 2026 Permalink | Reply 

      import istr

for f, l, o, w, e, r, p in istr.permutations(range(10), 7):
    if sum(istr("=flower")) ** p == istr("=flower"):
        for a, c, t in istr.permutations(range(10), 3):
            if istr("=flowerpact").all_distinct() and all(istr("=factor").is_divisible_by(x) for x in (a, r, t)):
                print(istr("=flower"), istr("=art"))

      Like

      • ruudvanderham's avatar

        ruudvanderham 11:32 am on 10 January 2026 Permalink | Reply

        This version is *much* faster. Note that it required the latest version of istr (1.1.16):

        import istr

for p in range(2, 10):
    for flower in istr.power_ofs(p, 100000, 1000000):
        if sum(flower) ** p == flower:
            istr.decompose(flower, "flower")
            for a, c, t in istr.permutations(set(istr.range(10))-set(flower)-{p}, 3):
                if istr("=flowerpact").all_distinct() and all(istr("=factor").is_divisible_by(x) for x in (a, r, t)):
                    print(istr("=flower"), istr("=art"))

        Like

    • GeoffR's avatar

      GeoffR 2:48 pm on 10 January 2026 Permalink | Reply

      Note: P = 5 is max value for int_pow() can accomodate without an error, but OK for a solution.

      % A Solution in MiniZinc
include "globals.mzn";

var 1..9:F; var 1..9:A; var 1..9:R; var 2..5:P;  
var 0..9:L; var 0..9:O; var 0..9:W; var 0..9:E;
var 1..9:T; var 0..9:C;

constraint all_different ([F, A, R, P, L, O, W, E, T, C]);

var 100000..999999:FLOWER == 100000*F + 10000*L + 1000*O
+ 100*W + 10*E + R;

var 100000..999999:FACTOR = 100000*F + 10000*A + 1000*C
+ 100*T + 10*O + R;

var 100..999:ART = 100*A + 10*R + T;

var 2..54:SUM = F + L + O + W + E + R;

constraint int_pow(SUM, P, FLOWER);

constraint FACTOR mod A == 0 /\ FACTOR mod R == 0 /\ FACTOR mod T == 0;

solve satisfy;
output["FLOWER = " ++ show(FLOWER) ++ ", ART = " ++ show(ART)];

% FLOWER = 390625, ART = 751
% ----------
% ==========

      Like

    • Brian Gladman's avatar

      Brian Gladman 7:40 pm on 10 January 2026 Permalink | Reply 

      from itertools import permutations

# convert digits to numbers and vice versa
dgts_to_nbr = lambda dgts: int(''.join(map(str, dgts)))
nbr_to_dgts = lambda nbr: tuple(int(d) for d in str(nbr))

# the digit sum of FLOWER is between the minimum 
# and maximum sums of six digits from 0..9
for sm in range(15, 40):
  # the possible powers to give sm ** P six digits
  for P in (4, 5):
    if 100_000 <= (FLOWER := sm ** P) < 1_000_000:
      t = (F, L, O, W, E, R) = nbr_to_dgts(FLOWER)
      # check that three digits remain unallocated
      if len(r3 := set(range(10)).difference(t + (P,))) == 3:
        # consider the order of the remaining digits for A, C and T
        for A, C, T in permutations(r3):
          FACTOR = dgts_to_nbr((F, A, C, T, O, R))
          # FACTOR must be divisible by A, R and T
          if FACTOR % (A * R * T) == 0:
            ART = dgts_to_nbr((A, R, T))
            print(f"FLOWER ART = {FLOWER} {ART} [{FACTOR = }, {P = }]")

      Like

  • Unknown's avatar

    Jim Randell 11:02 am on 6 January 2026 Permalink | Reply
    Tags:   

    Brain teaser 981: Bailing out 

    From The Sunday Times, 10th May 1981 [link]

    The pirate ship “Nancy” had one leaky hold. As long as the water was not allowed to rise above a certain level there was no danger of the ship sinking; so the practice was to allow the water to rise to a mark on the side of the hold which was just below the danger level, and then to send in a team of about 15 or 20 pirates to bale out the hold until it was empty. The water came in continuously at a constant: rate, so the process had, to be repeated regularly.

    There were two kinds of baling equipment: practical pans and capacious cans (which emptied the water at different rates), and each pirate who was assigned to baling duty picked up whichever of these was nearest to hand.

    One baling party consisting of 11 pirates with practical pans and 6 pirates with capacious cans emptied the hold in 56 minutes. When the water had risen again, the next party of 5 pirates with practical pans and 12 with capacious cans emptied it in 35 minutes. A third party of 15 with practical pans plus 4 with capacious cans took exactly an hour to do the job.

    How long would it take a party of 6 with practical pans and 10 with capacious cans?

    This puzzle is included in the book The Sunday Times Book of Brainteasers (1994).

    [teaser981]

     
    • Jim Randell's avatar

      Jim Randell 11:02 am on 6 January 2026 Permalink | Reply

      If we treat the volume of water at the point that baling starts as 1 unit, then we can write the following equation for a team with P pans and C cups as:

      T(P.p + C.c + f) = 1
      ⇒
      P.p + C.c + f = 1/T

      where:

      T is the total time taken to empty the hold (min);
      p is the rate at which a pan removes water (units/min);
      c is the rate at which a cup removes water (units/min);
      f is the rate at which the hold fills (units/min) (this will be negative as water is flowing into the hold).

      For the three teams given we get the following equations:

      11p + 6c + f = 1/56
      5p + 12c + f = 1/35
      15p + 4c + f = 1/60

      This gives us three equations in three variables, which can be solved to give the rates p, c and f.

      We can then calculate the time taken for the 4th team:

      T(6p + 10c + f) = 1
      ⇒
      T = 1/(6p + 10c + f)

      The following Python program runs in 74ms. (Internal runtime is 186µs).

      from enigma import (Matrix, Rational, printf)

Q = Rational()

# construct the equations
eqs = [
  #  p   c   f   1/time
  ((11,  6,  1), Q(1, 56)), # team 1 (11p + 6c) takes 56 minutes
  (( 5, 12,  1), Q(1, 35)), # team 2 (5p + 12c) takes 35 minutes
  ((15,  4,  1), Q(1, 60)), # team 3 (15p + 4c) takes 60 minutes
]

(p, c, f) = Matrix.linear(eqs, field=Q)
printf("[p={p} c={c} f={f}]")

# calculate time for team 4 (6p + 10c)
t = Q(1, 6*p + 10*c + f)
printf("team 4 takes {t} minutes")

      Solution: It would take the 4th team 42 minutes to empty the hold.

      We get:

      p = 1/840
      c = 1/336
      f = −11/840
      ⇒
      1/T = 6/840 + 10/336 − 11/840
      1/T = 1/42
      T = 42

      So the water is coming in at 11/840 units/min, and a pirate with a pail can empty out 1/840 units/min. So 11 pirates with pails would be able to remove the water at the rate it was coming in. A pirate with a cup can empty 1/336 units/min (so a cup can bail more than a pail), and 2.5 pirates with cups could remove water at the rate it is coming in.

      Like

  • Unknown's avatar

    Jim Randell 7:15 am on 4 January 2026 Permalink | Reply
    Tags:   

    Teaser 3302: Challenging grandson 

    From The Sunday Times, 4th January 2026 [link] [link]

    My grandson likes making puzzles; his most recent one started with a four-digit number with all the digits different. He added up all the possible two-digit combinations of these four digits (e.g., if the digits were 0, 1, 2 and 3 he would add 01 + 02 + 03 + 10 + 12 + …). This total divided exactly into his four-digit number.

    Not satisfied with this, he said he had added an extra digit (different from all the others) to the end of his number to make a five-digit number. The five-digit number was an exact multiple of the total of all the two-digit combinations of these five digits. He said that if he told me how many digits were in this multiple, I would be able to work out his five-digit number.

    What was his five-digit number?

    [teaser3302]

     
    • Jim Randell's avatar

      Jim Randell 7:35 am on 4 January 2026 Permalink | Reply

      When constructing the sum of the combinations, with 4 digits, each can be paired with 3 others, and so appears 3 times in the tens column and 3 times in the units column. The total sum is therefore:

      33 × sum(digits)

      And starting with 5 digits the total sum is:

      44 × sum(digits)

      This Python program runs in 68ms. (Internal runtime is 955µs).

      from enigma import (irange, nsplit, div, ulambda, filter_unique, ndigits, printf)

# generate possible scenarios
def generate():
  digits = set(irange(0, 9))

  # the first number is a 4-digit multiple of 33 with distinct digits
  for n1 in irange.round(1000, 9999, step=33, rnd='I'):
    ds = nsplit(n1)
    if len(set(ds)) != 4: continue
    # calculate the sum of the 2-digit combinations
    t1 = sum(ds)
    s1 = 33 * t1
    # n1 is a multiple of s1
    k1 = div(n1, s1)
    if k1 is None: continue

    # add in an extra digit to form the second number
    for d in digits.difference(ds):
      n2 = 10*n1 + d
      # calculate the sum of the 2-digit combinations
      s2 = 44 * (t1 + d)
      # n2 is a multiple of s2
      k2 = div(n2, s2)
      if k2 is None: continue

      # return (number, sum, multiple) values for each number
      printf("[n1={n1} s1={s1} k1={k1} -> n2={n2} s2={s2} k2={k2}]")
      yield ((n1, s1, k1), (n2, s2, k2))

# find solutions unique by number of digits in k2
f = ulambda("((n1, s1, k1), (n2, s2, k2)): ndigits(k2)")
ss = filter_unique(generate(), f).unique

# output solutions
for ((n1, s1, k1), (n2, s2, k2)) in ss:
  printf("n2={n2} [s2={s2} k2={k2}; n1={n1} s1={s1} k1={k1}]")

      Solution: The 5-digit number was 83160.

      There are 6 candidate scenarios (number = sum × multiple):

      2376 = 594 × 4 → 23760 = 792 × 30
      3564 = 594 × 6 → 35640 = 792 × 45
      4752 = 594 × 8 → 47520 = 792 × 60
      7128 = 594 × 12 → 71280 = 792 × 90
      8316 = 594 × 14 → 83160 = 792 × 105

      In each case the sum of the digits in the 4-digit and 5-digit numbers is 18, and so the sum of the 2-digit combinations from the 4-digit number is 33 × 18 = 594, and the sum of the 2-digit combinations from the 5-digit number is 44 × 18 = 792.

      Of the 5-digit numbers only the last one is a 3-digit multiple of the sum, and this provides the answer to the puzzle.

      Like

    • Ruud's avatar

      Ruud 8:51 am on 4 January 2026 Permalink | Reply 

      import peek
import istr
import collections

collect = collections.defaultdict(list)
for n4 in istr.range(1000, 10000):
    if not n4.all_distinct():
        continue
    s4 = 33 * sum(n4)
    if not n4.is_divisible_by(s4):
        continue
    for n1 in istr.range(10):
        if n1 in n4:
            continue
        n5 = n4 | n1
        s5 = 44 * sum(n5)
        if n5.is_divisible_by(s5):
            multiple = n5 / s5
            collect[len(multiple)].append(n5)

for n5s in collect.values():
    if len(n5s) == 1:
        peek(n5s[0])

      .

      Like

      • ruudvanderham's avatar

        ruudvanderham 7:15 pm on 4 January 2026 Permalink | Reply

        Slightly faster:

        import peek
import istr
import collections

collect = collections.defaultdict(list)
for p4 in istr.permutations(range(10), 4):
    n4 = istr.join(p4)
    s4 = 33 * sum(n4)
    if not n4.is_divisible_by(s4):
        continue
    for n1 in set(istr.range(10)) - set(p4):
        n5 = n4 | n1
        s5 = 44 * sum(n5)
        if n5.is_divisible_by(s5):
            multiple = n5 / s5
            collect[len(multiple)].append(n5)

for n5s in collect.values():
    if len(n5s) == 1:
        peek(n5s[0])

        Like

    • Frits's avatar

      Frits 11:43 am on 4 January 2026 Permalink | Reply 

      from itertools import permutations

# the extra digit must be zero due to the alternating sum rule
d = dict()
for A, B, C in permutations(range(1, 10), 3):
  # divisibility of 11 rule (alternating digit sum is 0 or a multiple of 11)
  if (D := (A + C - B) % 11) in {0, A, B, C, 10}: continue
  # calculate sum of digits (must be a multiple of 3)
  if (s := A + B + C + D) % 3: continue
  # calculate first multiple  
  m1, r = divmod(ABCD := 1000 * A + 100 * B + 10 * C + D, 33 * s)
  if r or m1 % 2: continue  # as m2 = 7.5 * m1
  # number of digits in second multiple must be unique
  d[nd2] = 0 if (nd2 := len(str(m2 := int(7.5 * m1)))) in d else 10 * ABCD
  
for k, v in d.items():
  if v:
    print("answer:", v)

      Like

      • Frits's avatar

        Frits 12:47 pm on 4 January 2026 Permalink | Reply 

        from itertools import permutations

# the extra digit must be zero due to the alternating sum rule
d = dict()
digits = set(range(1, 10))    
# D must be even as 10 * ABCD must be divisible by 44
for D in [2, 4, 6, 8]:
  for A, C in permutations(digits - {D}, 2):
    # divisibility of 11 rule (alternating digit sum is 0 or a multiple of 11)
    if (B := (A + C - D) % 11) in {0, A, C, D, 10}: continue
    # calculate sum of digits (must be a multiple of 3)
    if (s := A + B + C + D) % 3: continue
    # calculate first multiple  
    m1, r = divmod(ABCD := 1000 * A + 100 * B + 10 * C + D, 33 * s)
    if r or m1 % 2: continue  # as m2 = 7.5 * m1
    # number of digits in second multiple must be unique
    d[nd2] = 0 if (nd2 := len(str(int(7.5 * m1)))) in d else 10 * ABCD
      
for k, v in d.items():
  if v:
    print("answer:", v)

        Like

  • Unknown's avatar

    Jim Randell 11:16 am on 2 January 2026 Permalink | Reply
    Tags:   

    Teaser 2463: [Single-digit multiples] 

    From The Sunday Times, 6th December 2009 [link]

    I have used each of the digits 0 to 9 once each to make a one-digit number, a two-digit number, a three-digit number and a four-digit number. The fourth number is a single-digit multiple of the third; the third number is a single-digit multiple of the second; and the second is a single-digit multiple of the first.

    What is the four-digit number?

    This puzzle was originally published with no title.

    [teaser2463]

     
    • Jim Randell's avatar

      Jim Randell 11:17 am on 2 January 2026 Permalink | Reply

      Here is a solution using the [[ SubstitutedExpression ]] solver from the enigma.py library:

      It runs in 85ms. (Internal runtime of the generated code is 8.8ms).

      #! python3 -m enigma -rr

SubstitutedExpression

"div(GHIJ, DEF) < 10"
"div(DEF, BC) < 10"
"div(BC, A) < 10"

--answer="(A, BC, DEF, GHIJ)"

      Solution: The 4-digit number is 3402.

      We have:

      3402 = 6 × 567
      567 = 7 × 81
      81 = 9 × 9
      9

      And between them the numbers 9, 81, 567, 3402 use each of the digits 0-9 exactly once.

      Like

    • Ruud's avatar

      Ruud 2:21 pm on 2 January 2026 Permalink | Reply

      Here is recursive solution:

      import istr

def expand(n="", used=[]):
    for d in range(1, 10):
        next_n = d * istr(n if n else 1)
        next_used = used + [next_n]
        if len(next_n) == len(n) + 1 and istr("").join(next_used).all_distinct():
            if len(next_n) == 4:
                print(*next_used)
            else:
                expand(n=next_n, used=next_used)

expand()

      Like

    • GeoffR's avatar

      GeoffR 4:54 pm on 2 January 2026 Permalink | Reply 

      % A Solution in MiniZinc
include "globals.mzn";

var 1..9: a; var 0..9: b; var 0..9: c; var 0..9: d;
var 1..9: e; var 0..9: f; var 0..9: g;
var 1..9: h; var 0..9: i; var 1..9: j;

constraint all_different ([a, b, c, d, e, f, g, h, i, j]);

var 1000..9999: abcd == 1000*a + 100*b + 10*c + d;
var 100..999: efg == 100*e + 10*f + g;
var 10..99: hi == 10*h + i;
var 2..9: d1; var 2..9: d2; var 2..9: d3; 

% Four, three and two digut number constraints
constraint abcd div efg == d1 /\ d1 * efg == abcd;
constraint efg div hi == d2 /\ d2 * hi == efg;
constraint hi div j == d3 /\ d3 * j == hi;

solve satisfy;

output ["The four-digit number was " ++ show(abcd)
++ "\n" ++ "[a, b, c, d, e, f, g, h, i, j] = "
++ "\n" ++ show([a, b, c, d, e, f, g, h, i, j]) ];

% The four-digit number was 3402
% [a, b, c, d, e, f, g, h, i, j] = 
% [3, 4, 0, 2, 5, 6, 7, 8, 1, 9]
% ----------
% ==========

      Like

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