How does your mathematical brain work? Some people are able to visualise the answer to this problem, while others prefer to apply maths logic and rules based on facts they already know. Imagine a three dimensional version of noughts and crosses where two players take it in turn to place their 3D nought or cross onto the grid. The grid is made from 27 place holders arranged in a 3 x 3 x 3 array. The object of the game is to complete as many winning lines of three as possible. How many different winning lines of crosses are there? How many different ways of solving this problem can you think of? For an extra challenge, can you adapt a method to work out the number of winning lines in a 4×4×4 cube?
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Below are four possible ways of solving this conundrum. Can you make sense of each method? Laura says… I know that all winning lines must pass either:
- along an edge of the cube
- through the middle of a face
- through the centre of the cube
There are 12 edges on a cube so there are 12 winning lines along edges. There are 6 faces on a cube, and 4 winning lines that pass through the middle of each face, so there are 24 winning lines through the middle of faces. Finally we need to consider the winning lines that go through the centre cube: vertex to opposite vertex: 4 middle of edge to middle of opposite edge: 6 middle of face to middle of opposite face: 3 So, in total, there are 12+24+4+6+3=49 winning lines. _______________________________________________________ Kat says… Winning lines can either be:
- Not diagonal
My non-diagonal winning lines are: 9 from front to back, 9 from left to right, 9 from top to bottom. My diagonal winning lines: On each layer there are 2 diagonal winning lines so: There are 6 from front to back, 6 from top to bottom and 6 from left to right. There are 4 lines from a vertex to a diagonally opposite vertex. In total, there are 27+18+4=49 winning lines. ________________________________________________________ Rob says… There are 3 possible places where a line can start:
- at a vertex
- at the middle of an edge
- in the centre of a face
A cube has 8 vertices, 12 edges and 6 faces. From a vertex there are 7 other vertices that you can join to in order to make a winning line. 7×8=56 lines, but this counts each line from both ends, so there are 28 ‘vertex’ winning lines. From the middle of an edge there are 3 other middles-of-edges that you can join to in order to make a winning line. 3×12=36 lines, but this counts each line from both ends so there are 18 ‘middle of edge’ winning lines. From the centre of each face there is one winning line, joining to the opposite face, so there are 3 ‘centre of face’ winning lines. So in total, there are 28+18+3=49 winning lines. ______________________________________________________ Becky says… The winning lines can be counted by looking at lines:
- in each horizontal plane
- in each vertical plane from left to right
- in each vertical plane from front to back
- in the diagonal planes
On a plane there are 8 winning lines. In the cube, there are 3 horizontal planes, so 8×3=24 winning lines. There are also 3 vertical planes going from left to right, but now with only 5 new winning lines per plane, as the 3 horizontal lines have already been counted. So 5×3=15 winning lines. On the 3 vertical planes going from front to back, we now only have 2 new(diagonal) winning lines per plane. So 2×3=6 winning lines. Finally, there are also diagonal planes to consider. There are 4 winning lines going from corner to diagonally opposite corner. In total, there are 24+15+6+4=49 winning lines.