**Problem 5** **TRIPOLLO**
TRIPOLLO is a strategy game in which numbered marbles are placed in a rectangular array of holes on a board. As shown, there are three rows of holes. A player wins if the marbles are placed so that in each column, the sum of the numbers in the first two rows is three times the number in the third row.

(a) John has a 5-column board and marbles numbered 1 to 15.

So far he has placed them as shown in the diagram.

Show how he could successfully place the remaining marbles.

(b) With the same board and marbles. John made the following start:

Can he successfully complete this game? Give reasons.

(c) You have an 8-column board and marbles numbered 1 to 24. Show how you can win by placing all the marbles on the board.

(d) Show that with an 11-column board and marbles numbered 1 to 33 it is not possible to find a winning placement.

Explain the problem in as much detail as you can.