Because the money each sheep can sell is exactly the total number of sheep owned by the two brothers, the amount of money sold is a complete square number. According to the method of dividing money between them, the ten digits of the total amount of money must be odd, not even; And because the two brothers share the same amount of money, the single digits of the sum of the money must be even.

And the number of a complete square number can only be 0, 1, 4, 5, 6, 9, but the number of units at this time can only be 4 or 6 (it can't be 0, because after my brother took the whole 10 yuan, there was still less than 10 yuan).

When the unit number is 4, the ten digits of the corresponding complete square number are always even, not odd [this is because the complete square number with unit number 4 can only be a number in the form of 10n+2 or 10N+8, therefore, (10n+2) 2 =100n2.

When the unit number of a complete square number is 6, its corresponding ten digits are always odd, not even [this is because a complete square number with a unit number of 6 can only be a number in the form of 10n+4 or 10N+6, so (10n+4) 2 =16.

That is to say, the total amount of money that the two brothers sell sheep is a complete square number, and the ten digits of this complete square number are all odd, and the unit digit is always 6, so when the elder brother and younger brother take turns to take 10 yuan, there will always be 16 yuan left in the end; The two brothers share the same amount of money. In the end, they should each get 8 yuan, but my brother got 10 yuan, so the hat my brother gave my brother was worth 2 yuan money. For example, 42 = 16, 62 = 36, 142= 196, 162 = 256, ... all meet the conditions.