Space on base without a cube on it
Base grid layer 1 cube
layer 2 cube
3) The seven cubes on the first layer can make 8 different arrangements, as the space on this layer can be moved round the base to 8 different places, like in question 1.
Same as question 2 the second layer of 6 blocks can make 1 less arrangement than the first layer, as there is one less place for the space to move to. A block from the second layer can only be placed in an arrangement if there is a block from the first layer below it. There is always 1 less place you can put the space on the layer as you add layers to the tower. So the space on the second layer can be moved to 7 different places so making 7 different arrangements. To find the total number of combinations that can be made altogether multiply the number of combinations the first layer can make by the number the second layer can make.
8 x 7 = 56
So 56 is the total number of arrangements for these 2 layers.
4a) n
m
m = width n = length
mn = Area of base or number of squares on the base.
mn = number of places to put the space = number of arrangements that can be made
When there is 2 layers, layer1 has the same number of arrangements as the number of spaces on the base so number of combinations for the first layer equals mn (mn = base = number of places to put a space = number of combinations that can be made). As the second layer has one less block than the first and can only place blocks into the tower if a block from the first layer is underneath it meaning the space has one less place to move to so one less combination can be made so the number of combinations for the second layer equals mn – 1 (one less than the number of places on the base).
mn-1 = area of layer two
= number of places to put a space on layer two
= number of combinations that layer two can make
So then if you multiply the number of combinations layer one can make by the number layer two can you get the total number of combinations for two layers.
So the number of combinations that can be made with two layers with any size base equals:
mn(mn-1)
4b) because we have been told that every time you add a layer the new layer has one less block on it than the last so there is one less place to put a space meaning this layer can make one less combination than the last. So as I have found the first layer combinations equals m x n (equals the number of places on the base) and the second has one less so equals m x n – 1 (number of places on the base – 1) so I believe the equation for the third layers combinations equals m x n – 2 so to find out the number of combinations that can be made with three layers you must multiply these equations together, so mn(mn – 1)(mn – 2) equals number of combinations for three layers. So for each layer you go up you minus the extra block that is missing then multiply it with the number of combinations that were made of the layers before. e.g. number of combinations for six layers will equal:
mn(mn-1)(mn-2)(mn-3)(mn-4)(mn-5)
5.1) to work out these questions I will move one of the two spaces round the base to each space once (I will call this space the permanent = P) and each time I will see how many options I have to put the second space (these options will be marked with numbers) so these are the combinations for one layer:
5 + 4 + 3 +
(mn-1 + mn-2 + mn-3 +
2 + 1 = 15 combinations that can be made
mn-4 + mn-5) : as you can see I have used the equations from question 4 but instead of using them for each layer I have used them for every combination e.g. for combination one we can use all the places on the base except one (P) so the number of options available to place the second space equals mn – 1.
so if you add up all the options available to put the second space for each combination we find the total combinations for that layer so for this layer it is 5 + 4 + 3 + 2 + 1 = 15 combinations.
Key:
5.2) combinations for two layers
3 + 2 + 1 = 6
(mn – 3 + mn – 4 + mn – 5)
so 6 combinations can be made on the second layer for every one first layer combination. Now I must multiply the number of combinations one layer can make by the number the second layer can make (15 x 6 = 90) so all together the two layers can make 90 combinations.
5. 3) to work out this question I will do the same as I did in question 5. 1 and 5. 2 but with a bigger base. Combinations for one layer:
7 + 6 + 5
+ 4 + 3 + 2
+ 1 = 28 combinations for one layer on a two by four base
second layer combinations for one first layer combination:
5 + 4 + 3
+ 2 + 1 = 15 combinations for the second layer of a two by four base for one first layer combination.
Then once again to find out the total number of combinations that can be made by two layers on a two by four base we multiply the number of combinations that can be made on the first layer by the number that can be made on the second so: 28 x 15 = 420 = number of combinations that can be made altogether.
5. 4a) m
n
