Pocket Money

This problem actually doesn’t have a definite answer. Nowhere in the problem does it say how many days the two students work. If they work for less than 6 days then Sally will earn most money. If they work for more than 6 days, then David will get the most.

Mums Kitchen Floor

There are a large number of possible answers here. Each one can easily be checked to see that it has the right symmetries.

For the extension, colour the squares like a chessboard. When you remove two opposite squares you remove two squares of the same colour. So you have left 30 squares of one colour and 32 of the other. You can’t cover these with the combination tiles as each combination covers one square of each colour.

The Treasure Map

We have put all the information from the story on the map below.

				n
				n
		$$		n
				n

The square that the treasure is in is (3,3).

Robots

There are two possible ways for the red robot to move. It could go (1,1), (1,2), (1, 3), (2, 3), (3, 3), (3, 2), (3, 1), (2, 1) and back to (1,1). We’ll deal with the other case in a minute.

Similarly there are two ways for the blue robot to move. Suppose that it goes, go (4,4), (4,3), (4, 2), (3, 2), (2, 2), (2, 3), (2, 4), (3, 4) and back to (4,4). If you compare the different grid references you will see that the two robots are never on the same square at the same time.

But the blue robot might go (4,4), (3,4), (2, 4), (2, 3), (2, 2), (3, 2), (4, 2), (4, 3) and back to (4,4). In this case the two robots are on the same square at the same time twice – the squares (2,3) and (3,2).

If the red robot goes round its square the other way (anti-clockwise) it will meet up twice with the blue robot if the blue robot goes clockwise. Then they meet on the squares (3,2) and (2,3). If the blue robot insists on going anticlockwise, then they won’t meet up at all.

So the two robots can be made to simultaneously occupy the squares (2,3) and (3,2).

Carrie's First Cubes

1. To get three of its faces coloured, the small cubes have to be on the corners of the large cube. There are 8 corners to a cube, so 8 of the smaller cubes have three faces painted red.
2. To get two of its faces coloured, a small cube will have to be on the middle of an edge. In fact there is one per edge. As there are 12 edges of the large cube, there are 12 small cubes that have two faces coloured.
3. The cubes with one coloured face are in the middle of a face. There are 6 of these.
4. Those cubes that haven’t seen any paint must be in the centre of the large cube. There is only one such cube. This can be seen by first noting that there are 9 small cubes to a face. So there are 9 cubes on each of the three levels of the big cube to give a total of 27. Two opposite and parallel faces use up 18 different small cubes. The remaining faces on the outside of the cube are 3 + 1 + 3 + 1 around the ‘middle’ between the two opposite faces. Hence there are 18 + 8 = 26 small cubes on the outside of the large cube. Then there are 27 – 26 = 1 on the inside.

Noughts Solution

The first player can always win if she plays correctly. Call the first player, Player A and the second player, Player B. To win, Player A needs to put her first nought in the centre square of the board. Then, wherever Player B goes, Player A should copy that move but on the opposite side of the board. We show this in the picture below.

Player A has put the first nought in the centre (we have shown it as A1 to indicate that it was Player A’s first move). Then Player B has put a nought in the square marked B1. Player A has replied by putting her next nought in square A2, to match Player B’s move.

Now suppose that Player B has a move that will not mean that Player A can get three noughts in a row. Then there must be a square on the symmetrically opposite side of the board that is safe for Player A. If Player B doesn’t have a safe move, then Player A wins and doesn’t play symmetrically.