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.