Thought I'd check the arithmetic for 2 two-digit numbers, and it works!

I expect 41+14 to be 12 (two right plus two up equals two right and two up).

Long addition in long form below uses:

'=' to show equivalent lines (reordering of terms (1+2=2+1), spliting numbers (41=40+1), adding single digits (1+4=22))

'->' for when the algorithm gives a digit

'<' for when we move over a column

    41+14
    = (40+1)+(10+4)
    = 40 + 10 + (1+4)
    = 40 + 10 + 22
    -> 1s digit = 2
    < 4 + 1 + 2
    = 22 + 2
    = 20 + 2 + 2
    = 20 + 41
    -> 10s digit = 1
    < 2 + 4
    = 0
    -> done
    == 12

It still works in the alternative system (14+21 should equal 12)

    14+21
    =10+20+42
    ->2
    <1+2+4
    =13+4
    =10+3+4
    =10+31
    ->1
    <1+3
    =0
    ==12