S2T2

Jim Randell 8:26 am on 15 October 2019 Permalink Reply
Tags: by: J L Powell ( 2 )   

• 

  Jim Randell 8:27 am on 15 October 2019 Permalink | Reply

  The tribes on the island are described as “similar in appearance”, which I take to mean that the setter is not able to determine which tribe the person he is questioning is from, other than from the answers to the questions.

  This Python program runs in 84ms.

  Run: [ @replit ]

  from itertools import product
  from enigma import (subsets, join, printf)
  
  # honesty rating
  h = dict(R=2, B=1, G=0)
  
  # valid replies for tribe t, statement s
  def valid(t, s):
    s = bool(s)
    if t == "R": return s
    if t == "G": return not(s)
    return True
  
  # collect responses
  rs = list()
  
  # assign tribes
  for (t1, t2, t3) in subsets("RGB", size=len, select="P"):
  
    # check statement: "Is 2 more truthful than 3?" -> 1: "No"
    if not valid(t1, not (h[t2] > h[t3])): continue
  
    # choose a goal scorer
    for g in (0, 2):
  
      # check statement: "Who scored the winning goal?" -> 2: "2"
      if not valid(t2, g == 2): continue
  
      # assign houses
      for (h1, h2, h3) in subsets("RGB", size=len, select="P"):
        # no one lives in the same colour house as his tribe
        if t1 == h1 or t2 == h2 or t3 == h3: continue
  
        rs.append(((t1, t2, t3), (h1, h2, h3), g))
  
  
  # Q3 to player P: "Is the colour of your house Q?"
  for (P, Q) in product((1, 2, 3), "RGB"):
    i = P - 1 # index for player P
  
    # collect results
    (ys, ns) = (set(), set())
    for (tribes, houses, g) in rs:
  
      # solution is (house colours by tribe, goal scorer)
      s = (tuple(x + houses[tribes.index(x)] for x in "RGB"), ("2" if g == 2 else "not 2"))
  
      # answer is "Yes"
      if valid(tribes[i], houses[i] == Q):
        ys.add(s)
  
      # answer is "No"
      if valid(tribes[i], houses[i] != Q):
        ns.add(s)
  
    # examine the responses
    for (s, t) in zip([ys, ns], ["Yes", "No"]):
      # we only want one
      if len(s) != 1: continue
      (hs, g) = s.pop()
      # and the goal scorer must be determined
      if g.startswith("not "): continue
      # output solution
      printf("Q3 to player {P}: \"Is your house {Q}?\"; {t} -> houses = {hs}; scorer = {g}", hs=join(hs, sep=" "))
  

  Solution: Yes. (i) Redman in blue house. Blueman in green house. Greenman in red house; (ii) Redman scored the goal.

  The third question is asked of player 2, and is: “Is your house green?”.

  An answer of “No” allows the questioner to infer:

  Player 2 is Redman. His house is blue. He scored the goal.
  Player 1 and 3 are: Greenman in a red house and Blueman in a green house, but we don’t know which player has which number.

  Similarly, if the third question asked (still of player 2) is: “Is your house red?”, and the answer received is “Yes”, then we can infer:

  Player 2 is Greenman. His house is blue. He did not score the goal.
  Player 1 and 3 are: Redman in a green house and Blueman in a red house, but we don’t which player has which number.

  But we cannot determine who the goal scorer is, only that it isn’t player 2.

  ---

  If we consider the first question.

  If the person we ask is Blue, then they could answer anything. So the following are possible:

  #1=B, #2=R, #3=G
  #1=B, #2=G, #3=R

  If the person asked the first question is Red, then #2 does not tell the truth more often than #3:

  #1=R, #2=G, #3=B

  If the person asked the first question is Green, then #2 does tell the truth more often than #3:

  #1=G, #2=R, #3=B

  So there are four possible assignments of players to tribes, and #2 is not Blue, so we can ask them a question, and know we are not going to get a “random” answer.

  If we ask #2 “Do you live in a green house?”. We know if G is asked this they must answer “Yes” (as they don’t), so if we get an answer of “No” we know that #2 must be R, and they do not live in a green house.

  In this situation the houses are: RB, BG, GR, and we also know that R is #2, and so answered Q2 truthfully, so R scored the goal.

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