A bicycle shop has some number of unusual bicycles. All their bikes are identical (there’s more than one bike), and each bike’s front and back wheels has at least one spoke each (each bike has more than one spoke). You don’t know how many bicycles are in the shop, but you know there are between 200 and 300 spokes in total.

If you knew the exact number of spokes, you would be able to figure out the number of bicycles.

You don’t know the exact number of spokes, but just knowing the fact that you would be able to figure it out with that information allows you to deduce the answer. How many bicycles and spokes are there?

#### Solution

There are two different interpretations of the question (depending on whether you think the front and back wheels must have the same number of spokes or not), and luckily both of them result in a unique solution!

##### Assuming front and back wheels can have different numbers of spokes

17 bicycles with 17 spokes each.

The key is knowing you could figure out the number of bicycles if you knew the total number of spokes:

- For example, if the number of spokes was 220, there could be 11 bicycles with 20 spokes each or 20 bicycles with 11 spokes each. This means the number of spokes on each bike must be equal to the number of bikes (or in other words, the total number of spokes must be a perfect square).
- However, even if the number of spokes were a perfect square, let’s say 225, there could be 15 bicycles with 15 spokes each, or 5 bicycles with 45 spokes each. But since each bicycle has at least 2 spokes (two wheels with at least one spoke each), this ambiguity is resolved if the number of spokes per bike was a prime number.

17 is the only prime number whose square (289) is between 200 and 300.

##### Assuming front and back wheels must have the same number of spokes

11 bicycles with 22 spokes each (11 spokes on each wheel).

The same reasoning as above applies, except now the number of spokes on *each *wheel must equal the number of bikes, which must still be a prime number.

11 is the only prime number whose square (121), when doubled (242), is between 200 and 300.

Just curious, why wouldn’t we have 1 bicycle with 289 spokes?

The wording of the problem uses the plural “bicycles,” which implies more than one, but you’re right that this edge case is poorly specified. I’ve re-worded it to make that a little clearer. Thanks!