The Traversal Problem (3 dimensions)
[This problem is related to the (easier) 2-dimensional Diagonal Problem - click here for a full solution]
1. Imagine a cuboid made up of lots of little cubes.
2. Imagine a straight line (or “traversal”) connecting two opposite vertices of the cuboid
3. How many little cubes does the line pass through?
Note: Cubes are only counted if the traversal passes through them, so if your traversal goes exactly through a vertex or an edge separating cubes within the cuboid, then it is not considered to pass through the adjacent cubes. In the 2×3×4 example shown, the traversal actually passes through an edge in the centre of the cuboid.
Problem: How many cubes does a traversal pass through in a 60×120×156 cuboid? (Hint: Start small!)
(The solution to the 3d problem will be posted only on request. Have a go at it, but let me know if you want a full solution and I’ll just move it up my priority list.)
3 comments
[...] about the 3 dimensional version of this [...]
It’s got to be similar in that each block will be connected to 2 other blocks except for when passing through a node and at both ends of the line. so I recon start off with -2 for both ends then finding the nodes 90×120 = 30 nodes, 90×140 = 10, then for 3 dimensions 90+120+140-2-30-10=308?
You’re close Malley, I like what you’re doing with the -2 which considers the single node (or “vertex”) appearing at both ends of the line. But be careful as you may already have allowed for that. You’ve then considered the 90,120 pair and the 90,140 pair… what about the 120,140 pair?
You’re very close
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