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