S2T2

Jim Randell 9:41 am on 14 September 2021 Permalink Reply
Tags: by: Michael Fletcher ( 18 )   

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  Jim Randell 9:42 am on 14 September 2021 Permalink | Reply

  This Python program considers the number of members in each group, and performs the transfers to see if the conditions of the puzzle are met.

  It runs in 54ms.

  Run: [ @replit ]

  from enigma import irange, inf, printf
  
  # S = starting average weight of group A
  # K = number of transfers from A -> B
  # W = weight of transfer number K
  (S, K, W) = (100, 10, 100)
  
  # suppose each group has n members
  for n in irange(K + 1, inf):
    m = 2 * n
    printf("n={n} ({m} members)")
    # initial total weight of the group is...
    t0 = S * n
    # K members of the group are transferred
    for k in irange(1, K):
      # the total weight becomes
      t1 = (S - k) * (n - k)
      w = t0 - t1
      # average weights for group A and group B
      (a, b) = (S - k, w + n + k - 1)
      printf("{k}: transfer = {w}, A avg {a0} -> {a}, B avg {b0} -> {b}", a0=a + 1, b0=b + 1)
      t0 = t1
  
    if w == W:
      printf("*** SOLUTION: n={n} ({m} members) ***")
      break
    printf()
  

  Solution: There are 38 members in the club.

  The average weight in Group 1 decreases by 1 kg per week from 100 to 90, and the average weight in Group 2 decreases by 1 kg per week from 138 to 128. After 10 weeks there are 9 members in Group 1 and 29 members in Group 2.

  ---

  Analytically:

  If we suppose each group has n members, the the total weight of Group 1 starts out at 100n.

  After the first transfer the average weight of Group 1 is 99 kg, so the total weight is 99(n − 1).

  And the difference between these two totals accounts for the weight of the first person transferred:

  w[1] = 100n − 99(n − 1)
  w[1] = n + 99

  After second transfer the average weight of Group 1 is 98kg, so the weight of the second person transferred is:

  w[2] = 99(n − 1) − 98(n − 2)
  w[2] = n + 97

  In general at the kth transfer, we have:

  w[k] = (100 − k + 1)(n − k + 1) − (100 − k)(n − k)
  w[k] = n + 101 − 2k

  And we are told the weight of the 10th person transferred is 100 kg:

  w[10] = 100
  n + 101 − 20 = 100
  n = 19

  So, there are 2n = 38 members in total.

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