A natural number $N$ greater than 10 is called a "super-square" if the number formed by each pair of consecutive digits of the number $N$ (considered in the same order) is always a perfect square. For example, 8164 is a "super-square" because the numbers 81, 16, and 64 are perfect squares. Other examples of super-squares are 25 and 649.

a) How many "super-square" numbers exist?

b) What is the largest "super-square" number?
Solution

a) It will be useful later to observe the list of all two-digit perfect squares:

$$
16, \quad 25, \quad 36, \quad 49, \quad 64, \quad 81
$$

The number formed by the first two digits of a super-square must be one of the numbers in list (.1). Thus, let's start counting the super-squares from the number 16.

- The number 16 is a super-square. If we want to add a digit to the right of 16 to form a new super-square, it is necessary that this digit be a 4, because 64 is the only number in list (.1) starting with 6. Thus, we see that the only super-square of 3 digits starting with 16 is 164.

Since 49 is the only number in list (.1) starting with 4, the digit 9 is the only one that can be added to the right of 164 to form a new super-square.

Now, no number in list (.1) starts with 9, so it is not possible to create a new super-square by adding a digit to the right of 1649. We thus obtain that the list of super-squares starting with 16 is

$$
16, \quad 164, \quad 1649 \text{.}
$$

- As there are no numbers in list (.1) starting with the digit 5, the only super-square that starts with 25 is
- Following the same reasoning used above to find the super-squares that start with 16, we see that the super-squares starting with 36 are

$$
36, \quad 364, \quad 3649
$$

- Similar to the case of 25, the only super-square that has 49 as its first two digits is the number itself
- The super-squares starting with 64 are

$$
64, \quad 649
$$

- And finally, the super-squares starting with 81 are

$$
81, \quad 816, \quad 8164, \quad 81649
$$

Counting the super-squares obtained above in each of the cases, we conclude that there are 14 super-squares:

b) From the previous item, we have that the largest of all super-squares is the number 81649.

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