(related to Part: Tricks for Mental Maths)
You may ask somebody to
You "guess" the result, using your special mentalist power: \(2\)
The result is always 2, no matter which is the initial number thought by your partner. To prove it, you can just try all combinations or use the arithmetic operations another way round:
\(2+10=12\) \(12\cdot 3=36\) \(36 / 4= 9\) - this shows that the last three steps just serve for confusion, since the result of the sum of digits in the second step is always 9. To see this, there are the following possible combinations of a sum of digits:
\[0+9,\quad 1+8,\quad 2+7,\quad 3+6,\quad 4+5\]
In these cases, the possible results of the second step might be the numbers
\[9\text{ or }90,\quad 18\text{ or }81,\quad 27\text{ or }72,\quad 36\text{ or }63,\quad 45\text{ or }54.\]
Dividing all numbers delivers all combinations of possible numbers thought initially by your partner:
\[1\text{ or }10,\quad 2\text{ or }9,\quad 3\text{ or }8,\quad 4\text{ or }7,\quad 5\text{ or }6, \]
which are all numbers from \(1\) to \(10\).