TIME FOR A QUICK GAME 3rd year of secondary education

 1. CUTTING A CAKE Imagine that we have a cake with a rectangular base and that we want to cut it with a knife. We have to cut the cake in such a way that non of the cuts are parallel to each other, non of them are parallel to the cake base and at no time do all the cuts intersect at the same point. However, we have to try and cut the cake into the greatest number of pieces possible. In the window there is a rectangle which represents the cake as seen from above. Each of the segments that appear show where the cake has been cut. We can move these segments and change their length by moving the ends. 1.- Start "cutting" the cake, first by cutting it in two places, then three etc until you've used all the segments that appear on the electronic board. Draw a table in your exercise book to record the number times the cake is cut and the number of pieces the cake is cut into (you have to count the areas that are formed to do this). Remember that non of the cuts should be parallel to each other and when there are more than two cuts they should never all intersect at the same point. 2.- If you cut the cake 10 times how many pieces of cake would you get? What about if you cut it 16 or 30 times? 3.- Use this information you have obtained to work out an equation which allows us to predict the maximum number of pieces that can be made according to the number of times the cake is cut. 4.- If we draw n non-parallel straight lines on a sheet of paper where no more than two lines intersect at any one point, how many sections will be produced?

 2. ORDERING COUNTERS - 1 Imagine that you have some green and red counters like the ones from a game of ludo. There are an equal number of red and green counters and they are laid out in a straight line so that all the counters of the same colour are next to each other. In the game each move consists of swapping round the position of two consecutive counters. By the end of the game we have to make sure that each counter is next to another counter of a different colour, i.e. red, green, red, green, etc (or vice versa). We also have to try and do this in the fewest possible number of moves. In the three windows below we can see three lines containing 3, 4 and 5 counters of each colour respectively. As you change the value of m you will see different arrangements of the counters, which are in accordance with the rules of the game. The parameter stops increasing when the final arrangement is shown and the number tells us how many moves were needed to complete the game. (The order the moves are made in is one of many but the number of moves indicated is the lowest possible number).
 5.- Once you have seen all the possible arrangements on the electronic board draw a table in your exercise book to record the number of counters of each colour and the numbers of moves made. Afterwards, try drawing all the arrangements when there are 6 and 7 counters of each colour. It's a good idea to use a symbol to indicate each colour rather than having to colour all the counters different colours. It also makes it easier to correct any mistakes you make. 6.- If you had n counters of each colour how many moves would be the minimum number of moves required to arrange all the counters so that the colours alternate. 7.- What would be the minimum number of moves required to arrange the following number of counters so that their colours alternate in each case: 15, 24 and 50?

3. ORDERING COUNTERS - 2

In the following window there is another example but this time with 3 counters of each colour and three different colours. As before, when you change the value of m you will see different arrangements of the counters, which are in accordance with the rules of the game. Again. the parameter stops increasing when the final arrangement is shown and the number tells us how many moves were needed to complete the game. The initial arrangement is similar to the example above as all the counters of the same colour are laid next to each other.

 8.- Look carefully at the window and then use your observations to work out the minimum number of moves for n counters of each colour. Start by drawing the different arrangements for when we have four counters of each colour. 9.- Try and work out the minimum number of moves needed for n counters and m colours (Start by examining the example we have with four colours and gradually increase the number of counters of each colour). Josep Mª Navarro Canut ProyectoDescartes.org. Year 2013 