Counting to 1031 on two hands

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You can count to 31 on one hand and 1031 on two hands by using binary. Each finger has a value as a power of two

Right thumb = 2^0 Right index = 2^1 Right middle = 2^2 Right ring = 2^3 right pinky = 2^4 same on the left but 2^5-9 and add them all together

I think that you mean 1023. The 1031 is probably just a typo.

If you don't need 0, you could count to 1024 by reassigning 0 to be 1024.

If you do some other things such as include the direction the arms/hands are pointing. If you include arms as straight down, straight up, straight out and to the front, (each arm separately) you could add 4 more bits and count up to 2^14 (16,384) by not including zero or 2^14-1 (16,383) by including zero.

For example, arm positions could count as (binary) 00 - arm down, 01 - arm up, 10 - arm to side, 11 - arm in front. So left arm down, right arm up, 1023 on the fingers would be 00 01 11111 11111 (decimal 2047).

You may be able to use base three.
0 Finger/thumb outstretched
1 bent
2 touching hand

That's a nice extension to the problem.