## 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 *a* – *b* + 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.