The Lemon Fool

Posted: **January 29th, 2021, 11:57 am**

A large cube is formed by 27 smaller cubes, glued together, of uniform size. A termite starts at the centre of the face of any of the outside cubes and bores a path that takes her once through every small cube. This movement is always parallel to a side of the large cube, never diagonal.

Is it possible for the termite to bore her way through each of the 26 outside cubes once and once only, then finish her trip by entering the central cube for the first time? If possible, show how it can be done. If impossible, prove it.

Cinelli

Posted: **January 29th, 2021, 2:11 pm**

cinelli wrote:A large cube is formed by 27 smaller cubes, glued together, of uniform size. A termite starts at the centre of the face of any of the outside cubes and bores a path that takes her once through every small cube. This movement is always parallel to a side of the large cube, never diagonal.

Is it possible for the termite to bore her way through each of the 26 outside cubes once and once only, then finish her trip by entering the central cube for the first time? If possible, show how it can be done. If impossible, prove it.

Cinelli

Sorry, my trip to Königsberg got cancelled for covid.

Posted: **January 29th, 2021, 2:13 pm**

My gcse-age daughter provides the following solution after around 10 seconds thought:

[Spoiler]
The termite is eating a wooden rubik's cube, basically. The cube is laid on a table. The termite enters the centre cube of the top later then proceeds to an edge cube, then travels in a spiral, first all around the outside cubes of the top layer, then down and all around the outside cubes of the middle layer, leaving only the central cube, then similarly all around the bottom layer, finally proceeding to the middle cube of the bottom layer and finally going up to the central cube.

EDIT: Oh dear not quite right I fear! Can anyone spot the problem? Might have to lecture my girls!!

Who says girls can't think spatially?

GS

Posted: **January 29th, 2021, 2:19 pm**

GoSeigen wrote:My gcse-age daughter provides the following solution after around 10 seconds thought:

Who says girls can't think spatially?

GS

That involves a diagonal move near the end I think.

Posted: **January 29th, 2021, 2:21 pm**

UncleEbenezer wrote:
cinelli wrote:A large cube is formed by 27 smaller cubes, glued together, of uniform size. A termite starts at the centre of the face of any of the outside cubes and bores a path that takes her once through every small cube. This movement is always parallel to a side of the large cube, never diagonal.

Is it possible for the termite to bore her way through each of the 26 outside cubes once and once only, then finish her trip by entering the central cube for the first time? If possible, show how it can be done. If impossible, prove it.

Cinelli

Sorry, my trip to Königsberg got cancelled for covid.

I had the same idea before my daughter got involved. BTW You can go there by bike for exercise, but make sure you oil 'er nicely before you leave...

GS

Posted: **January 29th, 2021, 2:50 pm**

cinelli wrote:A large cube is formed by 27 smaller cubes, glued together, of uniform size. A termite starts at the centre of the face of any of the outside cubes and bores a path that takes her once through every small cube. This movement is always parallel to a side of the large cube, never diagonal.

Is it possible for the termite to bore her way through each of the 26 outside cubes once and once only, then finish her trip by entering the central cube for the first time? If possible, show how it can be done. If impossible, prove it.

Spoiler...

It's impossible. To see that, imagine the 27 cubelets are coloured white and black in a 3-dimensional analogue of a checkboard pattern, so that each of the 8 corner cubelets is coloured black, each of the 12 middle-of-an-edge-of-the-big-cube cubelets is coloured white, each of the 6 centre-of-a-face-of-the-big-cube cubelets is coloured black and the central cubelet is coloured white. There are 8+6 = 14 black cubelets and 12+1 = 13 white cubelets, and the termite's path moves to the opposite colour every time it enters a new cubelet. So to visit all 27 cubelets, the termite must both start and end in a black cubelet, and so it cannot end in the white central cubelet.

GoSeigen wrote:EDIT: Oh dear not quite right I fear! Can anyone spot the problem? Might have to lecture my girls!!

I'm afraid I spotted the problem almost as soon as I first saw the puzzle (which was at about 14:15), long before I saw your daughter's solution! It's a recurring theme in many puzzles, just in 3-dimensional form this time, and so I've seen it many times before.

But I would suggest trying to lead your daughter to find the problem for herself rather than lecturing... You'll be in a far better position to judge how best to go about that for your daughter than I am, but the sort of approach I would be thinking of is to say that it sounds a good idea, but like all good ideas, one should make certain whether it actually works by saying exactly what path the termite should follow...

Gengulphus

Posted: **January 29th, 2021, 3:16 pm**

GoSeigen wrote:Who says girls can't think spatially?
GS

Apart from your regular participation here, you're a Go enthusiast, yesno? Surely that offers some modest yardstick for your girls' spatial reasoning. Are they Dan-rated, as one might hope in someone who started early and is now around peak age for logical reasoning?

Posted: **January 29th, 2021, 4:47 pm**

Gengulphus wrote:But I would suggest trying to lead your daughter to find the problem for herself rather than lecturing... You'll be in a far better position to judge how best to go about that for your daughter than I am, but the sort of approach I would be thinking of is to say that it sounds a good idea, but like all good ideas, one should make certain whether it actually works by saying exactly what path the termite should follow...

Don't worry, the lecture comment was in jest, I do have the reputation with my kids of lecturing, but not for their failures, it's usually just a long monologue on one of my favourite topics: language, science, family history or sex. :shock:

We actually spread the contents of the recycling box over the kitchen floor and quickly located the flaw.

UncleEbenezer wrote:Apart from your regular participation here, you're a Go enthusiast, yesno? Surely that offers some modest yardstick for your girls' spatial reasoning. Are they Dan-rated, as one might hope in someone who started early and is now around peak age for logical reasoning?

Correct, Uncle, but I'm ashamed to say I've so far failed to infect them with my enthusiasm for the game. My older daughter is convinced she is a chess whiz having beaten me in an early game, and they know the rules of go but have thrown themselves into another Japanese pastime, Judo, which they both enjoy immensely. There's still time for the board games though.... I started in my 20s and learned pretty quickly because I was not afraid to study hard by that age.

GS

Posted: **January 30th, 2021, 12:18 pm**

GoSeigen wrote:My gcse-age daughter provides the following solution after around 10 seconds thought
GS

Good. Delighted that your daughter has had a go at this puzzle. I will bear her in mind for future brain teasers.

Well solved, Gengulphus.

Cinelli

The Lemon Fool

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