|
|
|
It's a Puzzlement
Self-Referential Aptitude Test
by Walter C. WrightThe following puzzle was created by Jim Propp, an associate professor in the department of mathematics at the University of Wisconsin at Madison. It is available on his Web page at www.math.wisc.edu/~propp/, and is used with permission. To solve the puzzle, it is useful to know that the answer to question 20 is E. In question 7, A and B are considered to be "1 apart," A and C to be "2 apart," etc. Without further ado, here is the puzzle as he presents it:
The solution to the following puzzle is unique; in some cases the knowledge that the solution is unique may actually give you a shortcut to finding the answer to a particular question, but it's possible to find the unique solution even without making use of the fact that the solution is unique. (Thanks to Andy Latto for bringing this subtlety to my attention.)
I should mention that if you don't agree with me about the answer to #20, you will get a different solution to the puzzle than the one I had in mind. But I should also mention that if you don't agree with me about the answer to #20, you are just plain wrong. :-)
You may now begin work.
1. The first question whose answer is B is question
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
2. The only two consecutive questions with identical answers are questions
(A) 6 and 7
(B) 7 and 8
(C) 8 and 9
(D) 9 and 10
(E) 10 and 11
3. The number of questions with the answer E is
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
4. The number of questions with the answer A is
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
5. The answer to this question is the same as the answer to question
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
6. The answer to question 17 is
(A) C
(B) D
(C) E
(D) none of the above
(E) all of the above
7. Alphabetically, the answer to this question and the answer to the following question are
(A) 4 apart
(B) 3 apart
(C) 2 apart
(D) 1 apart
(E) the same
8. The number of questions whose answers are vowels is
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
9. The next question with the same answer as this one is question
(A) 10
(B) 11
(C) 12
(D) 13
(E) 14
10. The answer to question 16 is
(A) D
(B) A
(C) E
(D) B
(E) C
11. The number of questions preceding this one with the answer B is
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
12. The number of questions whose answer is a consonant is
(A) an even number
(B) an odd number
(C) a perfect square
(D) a prime
(E) divisible by 5
13. The only odd-numbered problem with answer A is
(A) 9
(B) 11
(C) 13
(D) 15
(E) 17
14. The number of questions with answer D is
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
15. The answer to question 12 is
(A) A
(B) B
(C) C
(D) D
(E) E
16. The answer to question 10 is
(A) D
(B) C
(C) B
(D) A
(E) E
17. The answer to question 6 is
(A) C
(B) D
(C) E
(D) none of the above
(E) all of the above
18. The number of questions with answer A equals the number of questions with answer
(A) B
(B) C
(C) D
(D) E
(E) none of the above
19. The answer to this question is:
(A) A
(B) B
(C) C
(D) D
(E) E
20. Standardized test is to intelligence as barometer is to
(A) temperature (only)
(B) wind-velocity (only)
(C) latitude (only)
(D) longitude (only)
(E) temperature, wind-velocity, latitude, and longitude
Lights Out
Chris Yaure's solution to the problem of turning the lights out (Diagram A shows which lights are on) is, "There are four ways to turn out the lights. For purposes of my solutions, I have numbered the lights. The lights in the top row, from left to right, are numbered 1 through 5; the second row 6-10; down to the bottom row, which is numbered 21-25.
a. 3-7-12-14-19
b. 2-4-6-7-8-10-11-15-16-18-19-20-22-23-24
c. 1-5-6-7-8-10-12-14-16-18-19-20-21-23-25
d. 1-2-3-4-5-7-11-15-19-21-22-24-25
"Only five of the lights can be turned on without also turning on other lights. The lights are the center light (13) and the four lights at the inside corners (7-9-17-19)."
Diagram A X X X X X X X X X X Tom Struppeck observes that if any single bulb can be turned on, then any configuration can be lit. But there are as many configurations as there are ways to throw switches. Starting with no bulbs being lit, there are four nontrivial ways to throw switches that result in no bulbs being lit. So the mapping from the configuration of switches thrown to bulbs lit is not one-to-one. Thus there are some configurations that cannot be lit, and so there must be at least one bulb that cannot be lit by itself.
Diagram B X X X X X X X X X; X X X X X X X John Herder observes that each of the 25 switches controls 2 or 4 of the lights marked in Diagram B. This makes it impossible to turn on just one of the marked lights. David Uhland provided a similar diagram to show that a single light in a corner cannot be turned on. Mark positions 1, 2, 4, 5, 11, 12, 14, 15, 21, 22, 24, and 25 and apply a similar argument.
Alan Erlebacher, Paul Ivanovskis, Frank Karlinski, Mark Kertzner, Robert Muller, Leonard Myers, Matt Schultz, Sanford Squires, and W. Thomas Williams also submitted solutions.