Self-Reference Puzzles

As an undergraduate, I stumbled upon Jim Propp's Self-Referential Aptitude Test and spent a couple hours working on it. That experience stuck with me and I have since made smaller-scale versions and variants for students. This is a hard activity to pin down to an age group, but it can skew a little more abstract and occasionally arithmetic, so I'd peg it around 7th grade and up but have definitely had some luck with advanced elementary school children.

Setup

In each of these puzzles, the student fills out a quiz with some sort of multiple-choice format. The answer to each question impacts how the other questions can be answered, so the quiz can only really be answered correctly or incorrectly as a whole. The goal is to find a set of answers so that all of them are true simultaneously.

There are a few good sources for these online, including but not limited to:

• Jim Propp's original SRAT

• Martin Henz's SRQ and small SRAT

• A collection at Brainzilla

• A couple by Patrick Honner

The examples I'll furnish here chop the space of solutions down a bit, hopefully making them good low-floor toy problems to start with. You can find a running list here, but below are examples of each type. The T/F form is a little reminiscent of Knights and Knaves puzzles while the A/B/C form gives a little more room for experimentation.

T/F Quiz

Decide whether each statement is TRUE (T) or FALSE (F):

1.		The number of TRUE statements is odd.
2.		There are more TRUE statements than FALSE statements.
3.		The next two statements are TRUE.
4.		The next statement is FALSE.
5.		No two consecutive answers are the same.

A/B/C Quiz

Decide whether the answer to each question is A, B, or C:

1.		The next question with the same answer as this one is Question
		(A) 2     (B) 3     (C) 4
2.		The answer to Question 3 is
		(A) B     (B) A     (C) C
3.		The two answers that occur an equal number of times are
		(A) B and C     (B) A and C     (C) A and B
4.		An answer that occurs an even number of times is
		(A) A     (B) C     (C) B

This is a natural activity to have students collaborate on. A standard part of the deductive process should be explaining your reasoning. While most puzzle solvers are used to being their own sounding boards, having students be accountable to each other for their deductive leaps can be a good way to build that skill.

Initial Questions

While each of these puzzles might look like a unique beast, here are some basic focal questions:

• Are there any questions that you can answer or answers that you can rule out?

• Are there pairs (or small clusters) of questions that are tied together?

• Are there any questions where knowing the answer would help you answer a lot of other questions or rule out a lot of answers?

• When is guessing a good strategy?

Explorations

Here are some topics that might come up in the course of solving these puzzles:

• While deductively establishing the answer to a question or ruling an answer out is one approach, it is pretty easy to check whether a full set of answers is a solution. For example, the answer card ABCC cannot be a correct answer to the A/B/C Quiz above, and we see that pretty quickly by noting that answering #1 with A demands that we also answer #2 with A -- not B. The total number of possible answer cards to that quiz is

  \( 3^4 = 81 \)

  but maybe not all of them need to be checked. (For example, we can rule out not just ABCC, but any card that starts with AB or AC!)

• While I tend to create puzzles with a unique solution, that's not something these puzzles require. Once you've solved a puzzle, can you be certain that the solution you've found is the only solution?

• After students have solved a number of these puzzles, I usually challenge them to create their own. They can make them as short or long as they like and then trade with other students. While students are more likely to detect when a puzzle they've made has no solutions, the previously mentioned distinction of a puzzle have one or multiple solutions is a little more subtle. Can they create a puzzle that has exactly one solution?

Wrap-Up Questions

• What are some ways to get started again once you've gotten stuck?

• Are some ways of solving these puzzles better at assuring you that you've found the only solution than others? How and why?

• One way to make a puzzle is to create a list of questions first and then try to create answers that fit. What are some pros and cons of writing a puzzle this way?

• Another way to make a puzzle is to create an answer card first and then try to craft questions that they answer. What are some pros and cons of writing a puzzle this way?

Extensions

Since premade puzzles are in limited supply, the main extensions for this activity involve the design of new puzzles or finding novel ways to solve existing ones.

• If students are interested in programming, programming solvers for these sorts of puzzles can be fun. Since the spaces of answer cards are often relatively small, students can write simple brute force scripts. (This is much easier for the T/F format than the other multiple choice formats!)

• The real fun in designing your own puzzles is creating new ways to create constraints. Existing puzzles mostly have questions about which questions have a particular answer, which answers have a particular question, the number of questions a particular answer corresponds to, and things like that. The interesting parts are the novel ways to pose them in terms of things like proximity of one question to another, sums of question numbers, number families the question numbers belong to, and patterns in the answer sequence. Can you find a new way to make an answer to a question depend on the answers to others? Even one new idea in a puzzle can make it great!

• Switching the answer types from T/F to letters changes the ways that questions can be asked and answered. Using A/B/C/D/E, for example, opens up new ways to talk about answers (vowels versus consonants, runs in alphabetical order, forming words in the answer card, and so on). Instead of Questions 1-10 having answers A-E, you could write a puzzle with Questions A-J each with answers 1-5 could give new tricks for relating answers to one another (e.g., "The answer to this question is the sum of the answers to questions 3 and 4").

This post was sponsored by the Julia Robinson Mathematics Festival