Problem 1
To have the minimum number of coins, one pocket should have none. The next should have the next smallest number:1. 10 pockets in total is 0+1+2+3+4+5+6+7+8+9 which is 45.
Problem 2
The towers will have a total height of ∑n where n is the number of cubes. If ∑n is odd then it can’t be divided into 2 equal towers. ∑n will be even if n is a multiple of 4 or one less than a multiple of 4.
Problem 3
The sum of the first 2 of 4 consecutive numbers is always odd, because it is the sum of an odd and even number. The 3rd and 4th numbers total 4 more than the first 2. The final total will be the sum of the first 2 numbers doubled, which being an odd number doubled can never be a multiple of 4, add 4, which makes no difference as to whether it is a multiple of 4.