July 2017
Problem of the Month

The Game Devisers
by Virginia McCarthy

It was a slow day at the cafe when Eliza and Anna came up with a game to play.   Anna said,
"I'll think of a positive number of 75 or less.
Ask me yes-or-no questions and see how quickly you can guess the number."

Eliza, who had never been known to ask an irrelevant question, plunged in thusly:

1. Is it a prime number?
2. Is it divisible by 2?
3. Is it divisible by 3?
4. Is it divisible by 5?
5. Is it less than 25?

The questions are given in the order Eliza asked them.   After the fifth question was answered — and not before — Eliza had figured out Anna's number.

Can you determine the number and the answers that Anna gave to the five questions?

Solution to the Problem:

My apologies -- the question was poorly written.   It makes it sound like there is one answer, so many of us (including me) stopped when we found a combination that worked.   However, Guillaume St-laurent and Olivier Bouliane found another possible solution.   They wrote:
"We achieved 2 different solutions, which are : 1 and 15, through the following questions :
1 ) No, 2) No 3) Yes 4) Yes, 5) Yes — we can identify 15.
1 ) No 2) No 3) No 4) No 5) Yes — we can identify 1.
Which means there are 2 answers to this puzzle, and we cannot identify which of these numbers is the correct answer."

So, the correct answer to the problem that was posed is NO.   You cannot determine the number and the answers that Anna gave to the five questions. But I gave credit to anyone who sent in a number and answers to the five questions that produced that number, and extra credit to anyone who sent in multiple solutions.

After receiving Guillaume St-laurent and Olivier Bouliane's email, I decided to check all 32 possibilities and I found the following numbers that worked:
1, 15, and 49.

There are 32 possible ways that 
the five questions can be answered:

1. Yes Yes Yes Yes Yes
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      2
  (3) Is it divisible by 3?
      none / irrelevant question
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

2. Yes Yes Yes Yes No
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      2
  (3) Is it divisible by 3?
      none / irrelevant question
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

3. Yes Yes Yes No Yes
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      2
  (3) Is it divisible by 3?
      none / irrelevant question
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

4. Yes Yes Yes No No
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      2
  (3) Is it divisible by 3?
      none / irrelevant question
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

5. Yes Yes No Yes Yes
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      2
  (3) Is it divisible by 3?
      none / irrelevant question
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

6. Yes Yes No Yes No
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      2
  (3) Is it divisible by 3?
      none / irrelevant question
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

7. Yes Yes No No No
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      2
  (3) Is it divisible by 3?
      none / irrelevant question
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

8. Yes No Yes Yes Yes
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      3,5,7,11,13,17,19,23,29,31,
      37,41,43,47,53,59,61,67,71,
      73
  (3) Is it divisible by 3?
      3
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

9. Yes No Yes Yes No
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      3,5,7,11,13,17,19,23,29,31,
      37,41,43,47,53,59,61,67,71,
      73
  (3) Is it divisible by 3?
      3
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

10. Yes No Yes No No
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      3,5,7,11,13,17,19,23,29,31,
      37,41,43,47,53,59,61,67,71,
      73
  (3) Is it divisible by 3?
      3
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

11. Yes No No No No
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      3,5,7,11,13,17,19,23,29,31,
      37,41,43,47,53,59,61,67,71,
      73
  (3) Is it divisible by 3?
      5,7,11,13,17,19,23,29,31,37,
      41,43,47,53,59,61,67,71,73
  (4) Is it divisible by 5?
      7,11,13,17,19,23,29,31,37,
      41,43,47,53,59,61,67,71,73
  (5) Is it less than 25?
      29,31,37,41,43,47,53,59,61,
      67,71,73 -- cannot 
      be determined

12. Yes Yes No No Yes
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      2
  (3) Is it divisible by 3?
      none / irrelevant question
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

13. Yes No Yes No Yes
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      3,5,7,11,13,17,19,23,29,31,
      37,41,43,47,53,59,61,67,71,
      73
  (3) Is it divisible by 3?
      3
  (4) Is it divisible by 5?
      irrelevant question
  (5) Is it less than 25?
      irrelevant question

14. Yes No No Yes Yes
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      3,5,7,11,13,17,19,23,29,31,
      37,41,43,47,53,59,61,67,71,
      73
  (3) Is it divisible by 3?
      5,7,11,13,17,19,23,29,31,37,
      41,43,47,53,59,61,67,71,73
  (4) Is it divisible by 5?
      5
  (5) Is it less than 25?
      irrelevant question

15. Yes No No Yes No
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      3,5,7,11,13,17,19,23,29,31,
      37,41,43,47,53,59,61,67,71,
      73
  (3) Is it divisible by 3?
      5,7,11,13,17,19,23,29,31,37,
      41,43,47,53,59,61,67,71,73
  (4) Is it divisible by 5?
      5
  (5) Is it less than 25?
      irrelevant question

16. Yes No No No Yes
  (1) Is it prime?  
      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,
      71,73
  (2) Is it divisible by 2?
      3,5,7,11,13,17,19,23,29,31,
      37,41,43,47,53,59,61,67,71,
      73
  (3) Is it divisible by 3?
      5,7,11,13,17,19,23,29,31,37,
      41,43,47,53,59,61,67,71,73
  (4) Is it divisible by 5?
      7,11,13,17,19,23,29,31,37,
      41,43,47,53,59,61,67,71,73
  (5) Is it less than 25?
      7,11,13,17,19,23 -- cannot 
      be determined

17. No No No No No
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      1,9,15,21,25,27,33,35,39,
      45,49,51,55,57,63,65,69,75
  (3) Is it divisible by 3?
      1,25,35,49,55,65
  (4) Is it divisible by 5?
      1,49
  (5) Is it less than 25?
      49 -- THIS WORKS

18. No No No No Yes
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      1,9,15,21,25,27,33,35,39,
      45,49,51,55,57,63,65,69,75
  (3) Is it divisible by 3?
      1,25,35,49,55,65
  (4) Is it divisible by 5?
      1,49 
  (5) Is it less than 25?
      1 -- THIS WORKS

19. No No No Yes No
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      1,9,15,21,25,27,33,35,39,
      45,49,51,55,57,63,65,69,75
  (3) Is it divisible by 3?
      1,25,35,49,55,65
  (4) Is it divisible by 5?
      25,35,55,65
  (5) Is it less than 25?
      35,55,65 -- cannot be 
      determined

20. No No No Yes Yes
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      1,9,15,21,25,27,33,35,39,
      45,49,51,55,57,63,65,69,75
  (3) Is it divisible by 3?
      1,25,35,49,55,65
  (4) Is it divisible by 5?
      25,35,55,65
  (5) Is it less than 25?
      no answers -- irrelevant 
      question

21. No No Yes No No
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      1,9,15,21,25,27,33,35,39,
      45,49,51,55,57,63,65,69,75
  (3) Is it divisible by 3?
      9,15,21,27,33,39,45,51,57,
      63,69,75
  (4) Is it divisible by 5?
      9,21,27,33,39,51,57,63,69
  (5) Is it less than 25?
      27,33,39,51,57,63,69 -- 
      cannot be determined

22. No No Yes No Yes
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      1,9,15,21,25,27,33,35,39,
      45,49,51,55,57,63,65,69,75
  (3) Is it divisible by 3?
      9,15,21,27,33,39,45,51,57,
      63,69,75
  (4) Is it divisible by 5?
      9,21,27,33,39,51,57,63,69
  (5) Is it less than 25?
      9,21 -- cannot be determined

23. No No Yes Yes No
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      1,9,15,21,25,27,33,35,39,
      45,49,51,55,57,63,65,69,75
  (3) Is it divisible by 3?
      9,15,21,27,33,39,45,51,57,
      63,69,75
  (4) Is it divisible by 5?
      15,45,75
  (5) Is it less than 25?
      45,75 -- cannot be determined

24. No No Yes Yes Yes
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      1,9,15,21,25,27,33,35,39,
      45,49,51,55,57,63,65,69,75
  (3) Is it divisible by 3?
      9,15,21,27,33,39,45,51,57,
      63,69,75
  (4) Is it divisible by 5?
      15,45,75
  (5) Is it less than 25?
      15 -- THIS WORKS

25. No Yes No No No
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      4,6,8,10,12,14,16,18,20,22,
      24,26,28,30,32,34,36,38,40,
      42,44,46,48,50,52,54,56,58,
      60,62,64,66,68,70,72,74
  (3) Is it divisible by 3?
      4,8,10,14,16,20,22,26,28,32,
      34,38,40,44,46,50,52,56,58,
      62,64,68,70,74
  (4) Is it divisible by 5?
      4,8,14,16,22,26,28,32,34,38,
      44,46,52,56,58,62,64,68,74
  (5) Is it less than 25?
      26,28,32,34,38,44,46,52,56,
      58,62,64,68,74 -- cannot be 
      determined

26. No Yes No No Yes
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      4,6,8,10,12,14,16,18,20,22,
      24,26,28,30,32,34,36,38,40,
      42,44,46,48,50,52,54,56,58,
      60,62,64,66,68,70,72,74
  (3) Is it divisible by 3?
      4,8,10,14,16,20,22,26,28,32,
      34,38,40,44,46,50,52,56,58,
      62,64,68,70,74
  (4) Is it divisible by 5?
      4,8,14,16,22,26,28,32,34,38,
      44,46,52,56,58,62,64,68,74
  (5) Is it less than 25?
      4,8,14,16,22 -- cannot be 
      determined

27. No Yes No Yes No
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      4,6,8,10,12,14,16,18,20,22,
      24,26,28,30,32,34,36,38,40,
      42,44,46,48,50,52,54,56,58,
      60,62,64,66,68,70,72,74
  (3) Is it divisible by 3?
      4,8,10,14,16,20,22,26,28,32,
      34,38,40,44,46,50,52,56,58,
      62,64,68,70,74
  (4) Is it divisible by 5?
      10,20,40,50,70
  (5) Is it less than 25?
      40,50,70 -- cannot be 
      determined

28. No Yes No Yes Yes
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      4,6,8,10,12,14,16,18,20,22,
      24,26,28,30,32,34,36,38,40,
      42,44,46,48,50,52,54,56,58,
      60,62,64,66,68,70,72,74
  (3) Is it divisible by 3?
      4,8,10,14,16,20,22,26,28,32,
      34,38,40,44,46,50,52,56,58,
      62,64,68,70,74
  (4) Is it divisible by 5?
      10,20,40,50,70
  (5) Is it less than 25?
      10,20 -- cannot be determined

29. No Yes Yes No No
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      4,6,8,10,12,14,16,18,20,22,
      24,26,28,30,32,34,36,38,40,
      42,44,46,48,50,52,54,56,58,
      60,62,64,66,68,70,72,74
  (3) Is it divisible by 3?
      6,12,18,24,30,36,42,48,54,
      60,66,72
  (4) Is it divisible by 5?
      6,12,18,24,36,42,48,54,66,
      72
  (5) Is it less than 25?
      36,42,48,54,66,72 -- cannot 
      be determined

30. No Yes Yes No Yes
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      4,6,8,10,12,14,16,18,20,22,
      24,26,28,30,32,34,36,38,40,
      42,44,46,48,50,52,54,56,58,
      60,62,64,66,68,70,72,74
  (3) Is it divisible by 3?
      6,12,18,24,30,36,42,48,54,
      60,66,72
  (4) Is it divisible by 5?
      6,12,18,24,36,42,48,54,66,
      72
  (5) Is it less than 25?
      6,12,18,24 -- cannot be 
      determined

31. No Yes Yes Yes No
  (1) Is it prime?  
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      4,6,8,10,12,14,16,18,20,22,
      24,26,28,30,32,34,36,38,40,
      42,44,46,48,50,52,54,56,58,
      60,62,64,66,68,70,72,74
  (3) Is it divisible by 3?
      6,12,18,24,30,36,42,48,54,
      60,66,72
  (4) Is it divisible by 5?
      30, 60
  (5) Is it less than 25?
      irrelevant question

32. No Yes Yes Yes Yes
  (1) Is it prime?   
      1,4,6,8,9,10,12,14,15,16,18,
      20,21,22,24,25,26,27,28,30,
      32,33,34,35,36,38,39,40,42,
      44,45,46,48.49,50,51,52,54,
      55,56,57,58,60,62,63,64,65,
      66,68,69,70,72,74,75
  (2) Is it divisible by 2?
      4,6,8,10,12,14,16,18,20,22,
      24,26,28,30,32,34,36,38,40,
      42,44,46,48,50,52,54,56,58,
      60,62,64,66,68,70,72,74
  (3) Is it divisible by 3?
      6,12,18,24,30,36,42,48,54,
      60,66,72
  (4) Is it divisible by 5?
      30, 60
  (5) Is it less than 25?
      irrelevant question
 

Correctly solved by:

1. Guillaume St-laurent and Olivier Bouliane *	Cantley, Quebec, Canada
2. Ken Boone	Montreal, Quebec, Canada
3. James Alarie *	Flint, Michigan
4. Anna Vice and Eliza Sheffield	Tuscaloosa, Alabama
5. Brijesh Dave	Mumbai City, Maharashtra, India

* extra credit for finding multiple solutions!

Send any comments or questions to: David Pleacher