Find The Fox Puzzle

Adam Hrankowski

How long will it take you to corner the fox?

You have 5 holes and 1 fox. The holes are in a line, left to right, A, B, C, D and E. The fox is in one of these holes.
Fox image by Daniel Roberts / Pixabay (modified by Author)

You want to find the fox. Each morning you check one of the holes. (I have no idea why you are limited in this way.) If the fox is in that hole, you and he go off together for iced tea.

Each evening, the fox moves to a neighbouring hole, either one to the right or one to the left. For example, if the fox is in hole B on Sunday morning, then Monday morning, the fox may be in hole A or hole C. (I have no idea why the fox behaves this way.)

What search stategy will you use to ensure that you eventually find the fox? How many days (maximum) will your search take?

My Solution

Although the fox moves randomly, we will assume that he is trying to avoid you. (Hes shy.) We will further assume that he is a clever fox, and knows where you will look. (Shy and psychic.) Therefore, we will now ask: How can the fox avoid you for the longest time?

Follow this schedule.

SUNDAY: Check B. No fox. The fox could be in any of ACDE.

MONDAY: Check B. No fox. You now know that on Sunday, the fox could not have been in A. Otherwise the fox would have moved from there Sunday night to B. Thus, Sunday and today, the fox could be in any of CDE.

TUESDAY: Check C. No fox. The fox was not in B on Monday, so cannot be in A today. That leaves BDE for Tuesday.

WEDNESDAY: Check D. No fox. On Tuesday, the fox wasnt in E, or wed have it now. Last night, the fox could have moved from D to E, D to C, B to C or B to A. Today, the fox is somewhere among ACE.

THURSDAY: Check D. No fox. The fox must have been in A or C yesterday. The only place for it to have moved is now B.

We now know where the fox is. We need to corner that slippery little canine.

FRIDAY: Check C. No fox. The fox has moved from B to A.

SATURDAY: Check B. The fox is in B. Bring iced tea to share with the fox.

The puzzle assumes that the fox moves randomly. Therefore, in any of those days, the hole you check could be occupied. However, by Saturday, you will definitely have found your fox.

Is there more?

How does the puzzle work for different numbers of hole? A 1-hole puzzle isnt even a puzzle. There is nowhere for the fox to go. With 2 and 3 holes, we can find the fox by the second day. Four holes this begins to require some thought. (Try it!)

What about 6 holes? Is there a solution? Can we generalize to N holes? Infinite holes?

What if we place the holes in a ring? Or adjust the number of foxes?

Please share your thoughts below, or fire me off an email.

And enjoy your iced tea.

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I thought I was a renaissance man. Turns out I have ADHD.


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