The mechanical octopus is to light up all the glow orbs that forms the 8 corners of a cube. To do that it could use its laser pen and mirrors by passing a continuous beam of photons once through the center of every orb altogether to emit spectra of colors. If it can do it using as few as possible mirrors to light all 16 orbs of a 4x4 square grid, what setup shall the octopus do for the cube orbs?
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1$\begingroup$ In other words, find the minimum number of connected straight lines that pass through the 8 corners of a cube? Is it allowed for a line to hit a corner and then immediately change direction, or do all lines have to terminate strictly outside the cube? $\endgroup$– astralfenixDec 14, 2016 at 15:48
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1$\begingroup$ That can be allowed as long as the beam hit the center (small dot) once. $\endgroup$– TSLFDec 14, 2016 at 15:54
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1$\begingroup$ Nice sketch! +1 $\endgroup$– TechidiotDec 14, 2016 at 15:59
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1$\begingroup$ I don't think I understand what the 4x4 grid has to do with this. $\endgroup$– Gareth McCaughan ♦Dec 14, 2016 at 16:06
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1$\begingroup$ The mechanical octopus can light all the orbs on the cube, using the minimum number of mirrors that would be required to light all the orbs on a 4x4 planar grid. So you have to figure out how many mirrors are required to do the latter; then we know it can light the cube using that same number of mirrors. (And then, of course, the challenge is in showing how to do it with that many mirrors.) $\endgroup$– Rubio ♦Dec 14, 2016 at 17:52
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1 Answer
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The 4x4 grid can be lit up with
6 beams, ie 5 mirrors. This can be done multiple ways. Here is one:
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Thus we need to find a way of lighting up the cube with the same number of mirrors. My drawing isn't up to it, so I'll try and describe it, referring to the following letters:
First we have a beam through D and then B, extending out slightly past B (it doesn't matter how far). The second beam goes through F, stopping when we are directly below the centre of the cube. The third beam goes through G, stopping when we are level with the top of the cube. We can now go through C and A with the fourth beam. It is now easy to go through the final two (E and H) with two more beams, so we once again needed 6 beams, or 5 mirrors.
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$\begingroup$ On 4x4: The mirror on second orb top row blocks the beam . The beam hit the second orb at bottom row twice (overheat). On 1x1x1: All are glowing colorful. $\endgroup$– TSLFDec 14, 2016 at 18:55
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1$\begingroup$ @TSLF I've redone the 4x4 to avoid those issues. Given the constraints of the mirrors this was actually the hardest part, as you have a lot less freedom in 2D to avoid clashes. $\endgroup$– bobajobDec 15, 2016 at 10:53