## Brain teaser

Here’s a nice little puzzle I came across in Godfrey Thomson’s Instinct, Intelligence, and Character (1924):

“Imagine a cube, which is going to be cut in two by a straight saw cut. The saw-cut section, the raw face of the cut, can clearly be of various shapes, as square, or triangular (if a corner were cut off). How would you cut the cube so that the section may be a perfect plane hexagon?”

You are not to draw diagrams, or look at cubical or near-cubical objects while thinking it out.

If you are not a mathematician, your first thought (like mine) may be that it is impossible, or that it is a trick question, like those puzzles about making pyramids out of matchsticks. But accept the assurance that it is a genuine puzzle, with a straightforward solution.

Once you accept that there is a solution, it shouldn’t take too long to find it. But the interesting question is how you reach the solution. Do you get there purely by visual imagery, purely by logical reasoning, or some combination of the two? (Of course mathematicians may use analytical geometry or whatever, but I’m treating it primarily as a puzzle for the intelligent layman or -woman.)

I suspect that there would be considerable individual differences in approach, and that these might cast some light on different mental ‘factors’. Also, possibly, differences between men and women or groups with different genetic and/or cultural ancestry.

1. Eric Temple Bell used this as one of the tests he applied to supposed child geniuses that were brought to him. He was a Professor of Math at Cal Tech in addition to being a writer on popular math and a science fiction author (lurid 1930s stuff).

2. Partly analytical (ok, the cube has six sides), partly visual (can almost see that plane crossing the long diagonal), and have no way so far of proving whether I’ve got the right answer.

Are you sure that kinesthetic/practical (get cube, slice it) isn’t an interesting part of problem solving?

Here’s another little puzzle–if you cut a corner off a hypercube/tesseract, what shape do you get?
(IIRC, there’s an inaccurate definition of a tesseract in _A Wrinkle in Time_–just assume that you’re working with a four-dimensional cube.)

3. This is the kind of thing I’m pretty bad at. However, I think that it’s something that lots of video-gaming would improve skills on. Some time ago I read that test scores for kids were flat over 2 decades except for “spatial imagination” or whatever they called it, where there was a significant improvement, I I think that video games was the reason.

I’m so old I remember “Pong”, you know.

4. There was this study from earlier this year that found video games improved “visual perception”. I know I see Visual-spacial intelligence described as if one ability cluster, so maybe you’re right.

5. When does pi equal two?

6. OK, just to give a ‘solution’ to anyone who is still frustrated:

- take any face of the cube, and ‘draw’ a line (AB) joining the mid-points of two adjacent edges.

- take the opposite face of the cube and draw a similar line (CD) on that face diagonally opposite to the first line

- these two lines are equal and in the same plane. That plane intersects the 6 faces of the cube in 6 equal lines (including the 2 lines already drawn), which make 6 equal angles with each other.

Of course, for a full solution it is not sufficient to find the right construction, but to be able to prove (with reasonable rigour) that it has the properties required. This is arguably even more difficult, and I’m not sure I could do it to the satisfaction of a mathematician. But to prove one of the key points:

- ‘draw’ lines joining the diagonally opposite end-points of AB and CD to each other. From the way in which AB and CD were constructed, these end-points are the mid-points of opposite edges of the cube. The lines joining them therefore both pass through the centre of the cube. Two lines intersecting in a point uniquely determine a plane. Therefore AB and CD also fall in the same plane.

Without proving this, the ‘solution’ is just a plausible guess. Of course, there are other solutions, like Godless’s more sophisticated approach.

7. When does pi equal two?

Somewhere on the surface of a sphere (not going to figure it out all the way). Sort of like a triangle’s angles summing to between 180 and 360 degrees.

8. It seems godlesscapitalist (above) is
a thoughtless flatlander.

9. Allen, you’re on the right track.
It’s easy to see in your mind’s eye
so, don’t give up.

10. Pi = 2 at the equator of a sphere. Glad I didn’t give up.

11. I don’t understand what all this talk of pi taking one values other than 3.14… Can anyone explain what’s going here, or is this just a case of bad joking?

My thoughts on the cube. I haven’t thought about it much but I observed the following obvious (though not mathematically proven) things

Suppose the corners are (0,0,0)
(1,0,0)
….
(1,1,1)

A) There is exactly one hexagon-edge in each face of the cube.

B) Each such edge has its endpoints on two adjacent (intersecting) edges of the cube.

C)The angle between adjacent edges of the
hexagon is 120 degrees (because its a hexagon).

D) After we draw one putative edge with verticies 1 and 2, (I drew a cube btw), there is only one edge of the cube on which vertex 3 could lie if conditions A, B and C are to be satisfied. Apply the same principle to determine on which edges points 4, 5 and six lie.

Anyway that’s not a proof of anything. THose are just intutions which enabled me to sketch a hexagon cutting through the cube.

These vertices then seemed reasonable.

P1=(0,1/2,1)
P2=(1/2,0,1)
P3=(1,0,1/2)
P4=(1,1/2,0)
P5=(1/2,1,0)
P6=(0,1,1/2)

I think they are all on the plane given
by equation

2x+2y+2z=3

thus coplanar.

In short, my thinking process basically involved figuring out constraints then doing a constraints based search.
Hope I’m not wrong. It can be very embarassing when getting a math point wrong..haha.but I’m about to hit “Post” NOW…

12. Oops, I didn’t read (or I forgot about the the part) that said I coldn’t draw a diagram…. :( Oh well I am embarassed now.

BTW I do have a nice looking diagram.

13. Mostly by visual imagery.

Visualized a cube and plane cutting through it. Spun the cube around in my head until the plane was cutting through all six sides similarly.
Then I spun the cube on an axis perpendicular to the plane as a double check.
That axis fits directly through the two corners of the cube furthest from the plane, which would make sense.

The first comment amused me a little because when I was 15, my parents were a little concerned because I’d been “acting out” and generally being a teenager so they had me talk to a psychologist and he gave me a bunch of tests like IQ, MMPI, and some other stuff like that before telling my parents that there was nothing for them to worry about and that I was acting like a normal teenager.

However, on the IQ portion of the tests, there were two categories where I “went off the scale”, whatever the hell that means. One of those was 3d visualization. :)