Brainteaser 1734: Scratchcard
From The Sunday Times, 10th December 1995 [link]
Every Saturday The Daily Broadsheet gives readers a free scratchcard comprising two crosses and a number of ticks concealed beneath silver foil squares. Readers have to scratch just three of the squares to reveal what is beneath. Anyone who scratches only ticks can use the card to obtain a 50p discount on The Sunday Broadsheet. The probability of revealing three ticks, expressed as a percentage, is a whole number larger than 50%.
To increase circulation the paper intends to increase the number of silver squares, still with only two crosses. This will increase the chances of finding ticks when scratching three squares, and again the probability will be a whole number percentage.
How many more silver squares do they intend to add?
This puzzle is included in the book Brainteasers (2002). The puzzle text above is taken from the book.
[teaser1734]
S2T2 
Jim Randell 9:39 am on 17 October 2019 Permalink |
If there are n ticks (where n ≥ 3), then the probability of scratching off 3 ticks is:
This Python program finds values of n where this probability is an integer percentage, greater than 50%.
Run: [ @repl.it ]
from enigma import irange, inf, first, printf # find n, p where probability p is an integer percentage > m def solve(m): for n in irange(3, inf): (p, r) = divmod(100 * (n - 1) * (n - 2), (n + 1) * (n + 2)) if r == 0 and p > m: yield (n, p) if p == 99 and r > 0: break # find the first two (n, p) values with p > 50 ((n1, p1), (n2, p2)) = first(solve(50), 2) d = n2 - n1 printf("{d} more squares [{n1} ticks -> {p1}%; {n2} ticks -> {p2}%]")Solution: They intend to add 9 more squares.
With 16 squares (14 ticks and 2 crosses) the probability is 65%.
With 25 squares (23 ticks and 2 crosses) the probability is 77%.
In fact there are only 4 values for n which give an integer percentage for p:
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