# Street Number Puzzle

You were walking around on a major street with 5-digit street numbers (for example, 11213) and noticed one of the street numbers was special – any two consecutive digits formed a 2-digit perfect square (in the order the digits appeared in the original number).

What number was it?

#### Solution

The only possible solution is 81649. This is straightforward because there are only six 2-digit perfect squares and the solution needs to use four of them: 16, 25, 36, 49, 64, 81.

• We can eliminate 25 because no 2-digit square ends with 2 or starts with 5
• No 2-digit square ends with 3 or 8, so the solution must start with either 36 or 81, and we can eliminate the other 2-digit square (we cannot eliminate both or we will not have enough digits to form the 5-digit solution)
• So we know the solution consists of 16, 49, 64, and either 36 or 81
• For 16 to be part of the solution, either it follows a 2-digit square that ends in 1, or the solution must start with 16
• But the solution must start with 36 or 81, so 16 must follow a 2-digit square that ends in 1, which can only be 81
• There is only one possible order for these numbers: 81649