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.
Here is my brute force solution to the problem:
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!