The Great Elephant Race

2025-08-31

Riddle Cover

Problem

There are 900 elephants in a race. Bob and Charlie know which elephants are in 1st and 2nd place.

You are allowed to ask Bob a question about the race that he will respond to with a number from 1 to $n$ . After that Charlie will give you the number of either the 1st or 2nd place elephant.

You have to use Bob's answer to determine whether Charlie's number is the 1st or 2nd place elephant.

What is the lowest upper bound of $n$ Bob can give you so that you can determine if the elephant Charlie gave you finished in 1st or 2nd?

Yes, if you ask Bob who got first place, you can use his answer to determine whether the elephant Charlie gives placed 1st or 2nd. However the upper bound for $n$ in this case is 900 (there are better answers).

Can Bob answer with fractions or decimals?

No, he must either answer with whole numbers or have the ability to map his possible answers to whole numbers.

Can you ask Bob multiple questions?

You can make the question you ask Bob as long as you want. However, Bob is only allowed to answer with a single number. You are allowed to make each digit depend on a sub-question, but then the lowest upper bound would be the range of possibilities that Bob would answer with.

For example, you could ask Bob 2 yes or no questions and have him answer with a 2 digit number (1 digit per question) where 1 means yes and 2 means no. Since there would be 4 possible answers Bob could give (11,12,21,22), the upper bound for $n$ would be 4.

This Sounds Like a Math Problem?

What problem isn't a math problem! Here's a version of the riddle formatted as a math question:

Let $A$ = The set of answers Bob can give
Let $E$ = The set of unique elephants in the race
Let $\mathcal{F} = \{f : E \times E \times A \to \{0,1\}\}$ $F = {f : E \times E \times A \to {0, 1}}$ be the set of questions you can ask Bob where:
- $f(\text{🐘}_a, \text{🐘}_b, \text{ans}) = 1$ if elephants $\text{🐘}_a$ and $\text{🐘}_b$ can be distinguished using Bob's answer $\text{ans}$
- $f(\text{🐘}_a, \text{🐘}_b, \text{ans}) = 0$ otherwise

Find the minimum possible $|A|$ such that:

\begin{align*} &\forall \text{🐘}_a, \text{🐘}_b \in E, \\ &\exists \text{ans} \in A, \\ &\exists f \in \mathcal{F} : \\ &(\text{🐘}_a = \text{🐘}_b) \implies (f(\text{🐘}_a, \text{🐘}_b, \text{ans}) = 1) \end{align*}

Solution

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