Counting with Fingers

Note that from the problem, we have the finger has two states, either up (which we will call $1$ ) or down (which we will call $0$ ). Thus, we can count in binary using the $10$ fingers to get to $2^{10}-1=1023$ , without memorising any numbers.

The problem states that each finger is allowed to have an up or down state, which represents the on and off states in binary. For the first hand in binary, we reach 31, as this is the maximum value when all fingers re up - we can get this through adding the digits of the fingers, which are 1, 2, 4, 8, and 16 (taking the last number and doubling it. This is what allows for the encoding of Base-10 into Base-2). By adding these 5 numbers, we get:

$1+2+4+8+16=31$

For the larger sum of the two hands, all we have to do is continue this logic of doubling the last number to add to the total sum. This is shown below:

$1+2+4+8+16+32+64+128+256+512=1023$