Gross.
I guess sheer calculation power- but there must be a system.
You can group several combinations which always have the same result.
e.g. 13: 13-4=9
15: 15-6=9
Next one is 18, next one 27... see the system?
Ok, now why?
Take 23 and 32: one result is 18, the other one 27.
The 20.29s:
20: 2 kept steady. you always substract 2.
now oh wonder. 20 - 2 = 18.
add anything to 20, like 5: you also add it to what you substract. so even at 29 you calculate:
29 - 11 = 18.
so at the 30s you calculate:
3x-(3+x)
which is 3*10 + x*1 -3 -x = 30 -3 +x -x = 27.
wohoo, got the system.
so the last part is:
why is x*10 - x = 9*x?
well, except knowing the little x*9 and see it you can do prove it by complete induction:
assume that x*10 - x = 9 * x
start with x = 1:
1 * 10 - 1 = 10-1 = 9 = 9 * 1 True
continue with x-> x+1:
(x+1) *10 - (x+1) = (x+1)*9. True.
That means: this rule is true for all numbers!!!
E.g. be x = 1774:
1774 * 10 = 17740 - 1774 = 15966 = 9 * 1774
Hooray.
And the last line of my little proof explains also why:
10*x is x+x+x+x+x+x+x+x+x+x (10 times)
when you substract one x you get:
x+x+x+x+x+x+x+x+x (9 times) which is 9*x
:tongue: