Matches
 

Can you arrange seven matches so that each one touches the other six?

Alex

Are we talking about wooden matches or paper matches?

With two matches, place the heads together to form a "V" of about 60 degrees. Slide a third match between them so that all three heads touch.
[SPOILER] Repeat to form a second group. Place one group on top of the other at right angles. All six will touch the other five[/SPOILER]


[SPOILER]
Bend the last match in the middle and place the two ends on the junctions of the other match heads
[/SPOILER]

[QUOTE=philmoore]Are we talking about wooden matches or paper matches?[/QUOTE]
I don't think that it matters. They each have similar properties -- length, thickness, etc.

[SPOILER] and they can be bend without causing the two parts to separate [/SPOILER]

heh... leave it to mathematicans to spot loopholes in problem! :smile:

[U]Without[/U] bending or breaking any matches!

Phil, they have square cross-section, but I don't think it matters.

Alex

[QUOTE=akruppa]heh... leave it to mathematicans to spot loopholes in problem![/QUOTE]
What mathematican? I'm an Engineer!

[QUOTE]
[U]Without[/U] bending or breaking any matches![/QUOTE]
Spoil sport :)

[QUOTE=akruppa]heh... leave it to mathematicans to spot loopholes in problem! :smile:

[U]Without[/U] bending or breaking any matches!

Phil, they have square cross-section, but I don't think it matters.

Alex[/QUOTE]
There is a classic solution with 7 mutually touching right circular cylinders. It's essentially the one which Wacky gave, assuming that I read his description correctly.


Paul

If I understand Wacky's solution, he required that one match could be bent. Can it be done without bending?

Alex

[QUOTE=akruppa]If I understand Wacky's solution, he required that one match could be bent. Can it be done without bending?

Alex[/QUOTE]I misunderstood.

To be honest, I didn't read it properly.

Here's the classical solution which uses right circular cylinders:

[spoiler]Lay three cylinders flat on a surface such that their axes are at 120 degrees to
each other and the ends touching. Note that there will be a triangular hole
at the center of the shape.

Build another such shape with three more cylinders and place it on top of
the first three such that the six cylinders are at 60 degree spacings.

Slide a seventh cylinder down the (now hexagonal) hole in the center.[/spoiler]


Paul

Correct! I found this [URL=http://www.loria.fr/~zimmerma/problems/matches.html]image[/URL] of the solution on Paul Zimmermann's page.

Alex

[QUOTE=akruppa]Correct! I found this [URL=http://www.loria.fr/~zimmerma/problems/matches.html]image[/URL] of the solution on Paul Zimmermann's page.

Alex[/QUOTE]
Not quite the same as the classical solution, as Paul's puts thes ticking-out ends of the matches in contact too. However, it's the same basic idea.


Paul

[QUOTE=xilman]Not quite the same as the classical solution, as Paul's puts thes ticking-out ends of the matches in contact too. However, it's the same basic idea.[/QUOTE]

Not so sure. I think that's exactly the classic solution, apart from the fact that the classic one could rest on a table-top, so the vertical one doesn't extend downwards.

If the sticking-out ends don't touch, then that pair of matches doesn't touch, surely?

[QUOTE=fatphil]Not so sure. I think that's exactly the classic solution, apart from the fact that the classic one could rest on a table-top, so the vertical one doesn't extend downwards.

If the sticking-out ends don't touch, then that pair of matches doesn't touch, surely?[/QUOTE]Perhaps you're right.

I'm starting to get parity errors as my memory gets older.


Paul

[QUOTE=xilman]I'm starting to get parity errors as my memory gets older.
[/QUOTE]

It could be worse, you could be a socialite who has party errors. Or an economist with Pareto errors. Or an industrial chemist with purity errors!
Or...

[QUOTE=xilman] Lay three cylinders flat on a surface such that their axes are at 120 degrees to each other and the ends touching.[/QUOTE]

For the purpose of description, let's place the origin of a coordinate system at the center of your arrangement and align the z-axis to be perpendicular to the "surface". Now, we can futher specify that one of the cylinders has its axis parallel to the x-axis. One end is near the origin, and the other is a match length away in the +x direction.

Question: Does the axis of each cylinder (extended) pass through the origin?

As I understand your "classical" arrangement, the second layer is located in a plane where z = +1 diameter.
Further, one of those matches extends in the -x direction.

Question: Where does the +x match touch the -x match?
It appears to me that there is a gap.

[QUOTE=fatphil]It could be worse, you could be a socialite who has party errors. Or an economist with Pareto errors. Or an industrial chemist with purity errors!
Or...[/QUOTE]

A writer with parody errors

[QUOTE=Wacky][SPOILER]
Further, one of those matches extends in the -x direction.
[/SPOILER][/QUOTE]

is incorrect.

[QUOTE=fatphil]is incorrect.[/QUOTE]

But
[QUOTE=xilman] such that the six cylinders are at 60 degree spacings.[/QUOTE]

As a understand it, the only way to have 6 cylinders spaced at 60 degree intervals will require them to be in pairs where those two are 180 degrees apart.

Note:
I am not describing the solution (see PZ's diagram), but rather the arrangement described by PL (xilman).

There is also a second objection to this solution (Xilman’s, although I note that he does not claim it as his). The incircle of an equilateral triangle has of necessity diameter < the length of its sides. The length of its sides is the diameter of the match that is supposed to pass through this circle. Surely this is a contradiction that means that Xillman’s vertical match (the one on Wacky’s z-axis) can not be put in place as suggested.

The diagram here should make this quite clear.
[url]http://mathworld.wolfram.com/Incircle.html[/url]

Off topic: (or is it?)

I just checked for new posts and received this reply --

[QUOTE]vBulletin Message
Sorry - no matches.[/QUOTE]


:)

akruppa,

There is a better solution with matches that have square-cross section, as you assume.

Without loss of generality, we may assume the matches are actually cubes. (Really SHORT matches!) Then, in fact we can make 8 of them touch! Just stack them into one cube made up of 8 smaller cubes. They all touch at the corner in the center of the big cube.

Now, if you want them to touch along at least a 1-dimensional slice, that's a different story. But since, in the real world, *touching* is really interactions with the weak force...I think my solution is fine. :D

Hehe I wondered when someone would think of that.
Some riddles have very odd solutions that still work if you're change your assumptions. A cube may always have 6 sides, 8 corners, and 12 edges, but that doesn't mean all things called "cubes" are in fact actual cubes.
[spoiler] Ice cubes, or puns on cues, etc. [/spoiler]
Rectangles may in fact usually include squares. This may be annoying to people like me who were brought up thinking that rectangles may not have equal sides, but it's still a fact that technically, squares can also be considered rectangles. To be more accurate, I'd use the word quadrilateral.