Simple dissection: the equilateral triangle

[bonanova]

An equilateral triangle cannot be dissected into two new equilateral triangles, nor into three equilateral triangles.

Is there a highest number of equilateral triangles into which a single equilateral triangle cannot be dissected?

[Barcallica]

Isn't 1,4,9,16,25... only possible numbers?

[BobbyGo]

Isn't 1,4,9,16,25... only possible numbers?

For equilateral triangles of the same size, yes, I believe that is correct. But consider an equilateral triangle made up of various sized equilateral triangles. For example:

/\

/ \

/ \

/ \

/________\

/\ /\

/ \ / \

/____\ / \

/ \ / \ / \

/___\/___\/________\

[k-man]

and this number is 5.

Proof:

ET = Equilateral Triangle.

1) Any ET can be dissected into 4 ETs by connecting the middle points of its sides. Thus, if an ET can be dissected into n ETs then it can also be dissected into n+3 ETs by subdivision.

2) Any ET can be dissected into any even number k of ETs (k>4) by dividing one of its sides into k/2 equal segments and drawing line segments parallel to the other two sides forming a row of smaller ETs along the divided side.

From 1) and 2) any number greater than 5 is possible.

[bonanova]

Nicely done k-man.