# Make a topological torus-with-a-hole out of congruent squares that may share an edge or a vertex with other squares

Suppose we arrange, in 3-dimensional space, 8 identical solid cubes in space so they form a square-shaped ring (using a 3x3 arrangement of squares except for the one in the middle).

Its surface will be topologically a torus, composed of a total of 32 congruent squares, any two of which intersect each other along an entire common edge or a common vertex, or not at all.

If we now remove one of the 32 congruent squares, the 31 that remain will form a torus with a hole punched through it, and such that any two of the squares that intersect do so only along an entire common edge or a common vertex.

## Puzzle:

What is the smallest number of congruent squares that you can arrange in 3-dimensional space so that they form a topological torus with a hole punched through it, and such that any two squares that intersect each other do so along an entire common edge or along a common vertex?

As a torus with a hole is the same (topologically) as an

"8" shape

the answer should be

5

• A torus with a hole is homotopy equivalent to what people usually call a figure 8, but the latter is 1-dimensional whereas the torus with a hole is 2-. I don't see how to arrange five squares to get something homeomorphic to a torus with a hole; can you specify, please? Oct 31, 2021 at 4:25
• @msh210: A picture would be nice, but basically it is like this: rot13(Fgneg jvgu svir fdhnerf va gur cynar neenatrq nf n Terrx pebff. Sbyq gjb bccbfvgr nezf hc bhg bs gur cynar naq fgvpx gurve raqf gbtrgure gb znxr n gevnathyne ghor. Sbyq gur bgure gjb nezf qbjajneqf naq pbaarpg gurz gb znxr n frpbaq gevnathyne ghor.) Oct 31, 2021 at 4:55
• Thanks, @JaapScherphuis, couldn't have put it better myself. Oct 31, 2021 at 4:57
• @DanielAsimov, msh210 I find it fascinating that there appears to be such a stringent and universally accepted definition of the informal "8" shape which I made all up as I went along. Maybe it's a boldface 8 and even if not every single eight I have encountered so far had nonvanishing thickness. It's all moot anyway since we are asked to work with solid squares in the end. Oct 31, 2021 at 18:04
• A "figure-8" in topology almost always means the space created by glueing two circles together along a common point. It's not a technical term, but it's almost always interpreted this way. Nov 1, 2021 at 3:22