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Find out how many different combinations 5 cubes can make on top of a 2 by 3 base.

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Introduction

Max Holloway Maths GCSE CW

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  1. I want to find out how many different combinations 5 cubes can make on top of a 2 by 3 base

Because the 5 cubes are on top of a base with 6 spaces on it there will always be one place free on the base (with out a cube on it) so this space can be moved around the base thus rearranging the cubes so creating different combinations.

Because there are only 6 spaces on the base (refer to Fig1) and only one space without a cube on it, this space can only be moved to 6 different spaces (refer to fig2) so making six different arrangements. So the number of combinations that can be made is limited by the number of places the space can be moved to so as there are 6 spaces on the base, 6pplaces to put a space, 6 combinations can be made (number of places to put a space = number of combinations that can be made)

Fig1: The area of the base equals the number of spaces.                                                                                                            

                                                           3

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2 X 3 =6                           2

Fig2: SP= space without cube on it. The space can be moved

round to 6 different places so 6 different arrangements can

be made

(space can be moved to any of the spaces on the base to make a different combinations)image08.png

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Fig 3: The 6 combinations

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Middle

                                                         layer 2 cubeimage01.pngimage16.pngimage12.pngimage13.pngimage14.pngimage15.png

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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 image05.png

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                 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

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Conclusion

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:image06.png

5.2) combinations for two layers

image25.pngimage23.pngimage24.png

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                                    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:

image27.pngimage29.pngimage26.png

                            7                   +                   6                  +                   5

image32.pngimage31.pngimage30.png

+                      4                  +                   3                  +                   2

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 +                    1   = 28 combinations for one layer on a two by four base

second layer combinations for one first layer combination:

image36.pngimage35.pngimage34.png

                        5                  +                   4                  +                  3

image38.pngimage37.png

+                     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

image17.png

                n

Max Holloway

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