## Challenge 233: What's left?

a        The numbers 1, 2, 3, ... 19, 20 are written on a classroom board.

The children in a class take it in turns to rub out two of the numbers, say a and b, and write instead the number a + b – 1.

What number will be written on the board when only one number remains?

b       The numbers 1, 2, 3, ...19, 20 are written on a classroom board.

The children in a class take it in turns to rub out two of the numbers, say a and b, and write instead the number ab + a + b.

What number will be written on the board when only one number remains?

c        The numbers 1, 2, 3, ... 19, 20 are written on a classroom board.

The children in a class take it in turns to rub out two of the numbers, say a and b, with a the larger and b the smaller of the two, and write instead the number ab + 1.

Show that

i      the number written on the board when only one number remains cannot be even;

ii     any odd number can be written on the board when only one number remains.