5=10 using the following logic.
Switch to the binary system of counting: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010 represent the numbers 1 to 10 decimal in binary. If we cross out those numbers with an odd number of ones we get:
11, 101, 110, 1001, 1010. When these are converted back to decimal we get: 3, 5, 6, 9, 10.
These are the listed numbers in order: so 1=3, 2=5, 3=6, 4=9 and 5=10.
Another way of solving the problem is to list the numbers in order that are the sum of an even number of powers of 2; or the sum of a pair of powers of 2.
(1) 3=2+2^0; (2) 5=2^2+2^0; (3) 6=2^2+2^1; (4) 9=2^3+2^0; (5) 10=2^3+2^1; (6) 12=2^3+2^2, etc.