Immensely Questionable Tests
I have finally become a queen — though only in a research paper. In Immensely Questionable Tests, I am Queen Tanya, and my ten PRIMES STEP student coauthors are my sages.
The project began with a probability puzzle from Mathematical Puzzles and Curiosities, my book with Ivo David and Yogev Shpilman. Here I present an equivalent version, without probabilities, to match our project.
Puzzle. How many correct answer choices are there?
a) 1 b) 1 c) 2 d) 2
The intended answer is 2: exactly two choices say 2. So far, so good. Now erase one of the 1s. How many correct answer choices are among a) 1, b) 2, and c) 2?
Now 1 appears once and 2 appears twice. Either value can be an answer. But calling all three choices correct would require the answer to be 3. What is going on here?
We restrict the answer choices to positive integers. We call a value m valid if it occurs exactly m times among the choices. Now we can discuss valid answers without getting into paradoxes.
Our first paper Self-Referential Tests counts tests with valid answers as a function of the cost, defined as the sum of all test options. The paper answers many questions related to different types of tests. For example, what is the cheapest test with k distinct valid values? The cheapest such test uses one 1, two 2s, and so on, through k copies of k. Its cost is 12 + 22 + … + k2. For three distinct valid values, the cheapest test is 1, 2, 2, 3, 3, 3, costing 14.
Our second paper, Immensely Questionable Tests, mentioned at the very start of this essay, deals with the paradoxical nature of such tests.
For example, consider a test with these options:
1, 2, 2, 3, 3, 3, 3, 4, 4, 4.
The valid values are 1 and 2, accounting for three options. This tempts us to say that the correct answer is 3. But there are four threes on the test, suggesting that the correct answer should be 4 instead. And there are three fours, taking us back to 3. We are going in circles! It is surprisingly difficult to decide which answers should be called correct. We were not able to agree on the answer. So we turned our confusion into a story about a queen and her sages. Ironically, even in a fairy tale, I keep teaching people mathematics.
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