Matches 2

[jfgw]

Take six identical matches and arrange them so that every match touches every other match. You are not allowed to break them.


Julian F. G. W.

[ReformedCharacter]

Take six identical matches and arrange them so that every match touches every other match. You are not allowed to break them.


Julian F. G. W.

Does 'touches every other match' mean touches all the other matches or touches each alternate match in a series?

RC

[jfgw]

Does 'touches every other match' mean touches all the other matches or touches each alternate match in a series?

That is an excellent point and would have made a good trick question. It is not a trick, however. Every match touches all five of the other matches.


Julian F. G. W.

[kempiejon]

I think I can see this. I tried with some tooth picks but it was too fiddly. It's 2 sets of three matches each making a triangle. A little triangle with long ends in the centre of a large triangle with neat vertices. The triangles share centres and are arranged so the little triangle's long edges bisect the vertices and opposite edges of the larger.

[jfgw]

I think I can see this. I tried with some tooth picks but it was too fiddly. It's 2 sets of three matches each making a triangle. A little triangle with long ends in the centre of a large triangle with neat vertices. The triangles share centres and are arranged so the little triangle's long edges bisect the vertices and opposite edges of the larger.

I think something like that could work but you would have to bend the matches. You would have to cross the opposite edges off-centre, however.

I found some coloured plastic bendy sticks which seem to work,

I'm not sure you could do it with normal matches without them breaking though. You could try those long cook's matches, or the cardboard ones out of a book.

There is a way to do it without bending the matches, however.


Julian F. G. W.

[kempiejon]

There is a way to do it without bending the matches, however.

That was what around about what I had in mind.
I'll think on for another version.

[9873210]

Take six identical matches and arrange them so that every match touches every other match. You are not allowed to break them.


Julian F. G. W.

Arrange three in a narrow arrowhead on a plane. The outside two touch at the point of the arrow and the tip of the third touches the sides of the outer two about a cm back from the point of the arrow. Put the other three in a similar formation and on top of and perpendicular to the first three.

[jfgw]

Correct answer 9873210.

Spoiler link,

https://i.imgur.com/xy1EhgW.jpg


Julian F. G. W.

[9873210]

I think I can see this. I tried with some tooth picks but it was too fiddly.

If you want to continue to do physical experiments, you might try using blu tack or modelling clay to hold things in place.

[kempiejon]

I think I can see this. I tried with some tooth picks but it was too fiddly.

If you want to continue to do physical experiments, you might try using blu tack or modelling clay to hold things in place.

Good idea re sticking together - cocktail sticks are not good replacements for matches either. I came to post my solution found with a bit more fiddling.
Unlike the flat solution 9873210 posted with 2 layers of 3 offered I had a vertical match. In the gap formed by the 3 on the bottom layer I had a match pointing up then only 2 matches with the point in the opposite direction surrounding it. I see now it's a more complicated answer though.

[9873210]

A 2x2x2 stack of cubes all touch at the single point at the center. This can be extended to eight rectangular prisms, so butt ends of eight matches might be arranged this way and you can delete any two to solve the problem.

However this only work if the adjoining corners of the matches are perfectly pointed. Any rounding and they will not touch at the point. In the solution above not only do the matches touch on at least a line but with a slight variation* there is enough wiggle room to accommodate small imperfections in any of the matches.

*On one arrow, pull one of the outside matches past the point, and have it's butt end contact the side of the one of the matches in the other layer. This may be needed because with imperfect matches the nine points of contact between the layers may not be coplanar, and with certain imperfections some of the matches may not touch.

[dealtn]

In a 2 dimensional plan it is easy to see how a simple "cross" solves how 4 matches can be arranged so each match touches each of the 3 others.

Extending this idea to 3 dimensional space, adding an additional match to each of the "front" and "back" of that cross, shows how 6 matches all meet in the "middle", thus touching 5 others.

The physicality of the round heads of the matches might prove a limit to how even small objects can't quite touch, but mathematically conceptually this is straightforward.

[jfgw]

Extending this idea to 3 dimensional space, adding an additional match to each of the "front" and "back" of that cross, shows how 6 matches all meet in the "middle", thus touching 5 others.

I can see how it would work with four matches in two dimensions.
They could be arranged slightly offset so that opposite matches touched at the corners. A fifth match could be placed at 90 degrees to the others so its end touched the other four. One pointing downward would not, however, touch the upward-pointing one.

This could be modified, however. Arrange the four matches to make a cross as before. Place one end of another match under the centre of the cross at 45 degrees to the others. You may want to prop up the ends of the cross so that it remains level. Now place another match on top directly above the bottom one.

You will need some clamps or weights to hold the top one down as it will need to bend slightly to contact all of the matches in the cross in the middle, plus the other end of the bottom match.


Julian F. G. W.