### Vn128.1 **Robby’s Topological Game** 4/2/78

About a month ago, Robby was shown a paper-cutting game by a

classmate’s parent. The procedure to follow was this:

1. Cut two paper strips of equal length (8″ will do)

2. Draw a line down the middle of each (using lined paper makes

this unnecessary)

3. Bend each strip of paper into a circle and tape the juncture

4. Join the circles perpendicularly and tape the juncture

5. Cut around the mid-line of each circle.

When two strips of equal length are so connected and cut, the surprising

result is that, though having passed through a circular phase, the strip

halves end up taped together as a square.

Robby enjoyed this game when shown it. Yesterday, I removed

a paper form he had made in the past (an 8 x 11 sheet divided into 11

strips 8″ long) from my clipboard and gave it to him. When I inter-

rupted his reading to give him this sheet of paper, Robby recalled the

game and quietly took it up on his own. He was very happy when the

procedure produced a square and showed it to Gretchen and me. We neither

paid much attention.

Going on to three circles, Robby cut two of the three along

their mid-lines. He judged (in error) that he had finished by finding

a square with a bar (a double strip) across the center. It lay flat.

Still no one paid attention. Robby went on to four circles, and he

cut all the mid-lines. What he got was a confusion of floppy paper.

I advised him to try to get it lying flat. Robby again borrowed my

clipboard, clipping and taping the product to it. He was delighted

when he succeeded in flattening the strip-figure and subsequently

taped it to a large piece of cardboard. The resulting shape is this:

But why stop at 4? Robby went on to connect and cut 5 circles. Here

he met another surprise. When cut, the 5 circles separated into

identical, non-planar shapes. Robby likewise taped these to another

piece of cardboard. When he made a cutting of 6 circles, controlling

the floppy strip-figures became a big problem. Robby succeeded at

taping it to the box from which he had been cutting cardboard backing

pieces, but in doing so went over an edge. He decided the problem

was getting too complicated to be fun and quit.

This morning I told him I had been thinking about his paper

cutting game and asked Robby to find the figure made from three circles.

When he returned, I asked him if he had cut all three circles. Robby

thought so, but when I pointed out the middle bar in his square was

double thick, he agreed he had only cut two. Robby saw immediately

that his square would divide into two rectangles. He cut the center

strip. “The 5’s made 2 too. Hey! I’ve got a new theory: the odd-

numbered circles make 2 and the evens all stay together.” I agreed

that this was an interesting speculation and that I could believe it

might be true, but that I couldn’t see immediately why it should be.

**Relevance**

I see this incident as one exceptionally valuable for

characterizing how significant learning occurs very naturally in a

mildly supportive milieu. First note that the initial exposure to the

“phenomenon” was quite memorable and puzzling. (How can you make a

square from two circles?) Robby clearly marked this phenomenon in his

mind as one which he would explore later. This pending explorarion was

invoked by the accident of his seeing a piece of paper approximately

meeting the material requirements for use in the game. The circum-

stance was one of no pressure. (He had been reading all of Gretchen’s

collection of Oz books and was probably a little bored.) He had no

outside direction or motivation at all. Once Robby succeeded at

making a square, he continued executing the procedure with stepwise

complications all focussed on one variable — the number of circles.

(He might have chosen to make the strips of different lengths — a

possibility he mentioned.) With the 3 circles, Robby stopped prema-

turely because he had produced a result (a square with a bar) only a

little different from the next simpler case (2 circles make a square).

With 4 circles, the outcomes of cutting were apparently sufficiently

confusing that completion could not be judged from the product but

depended on verifying that individual steps of the procedure were

completed. With the figure of 4 circles he was excited and delighted

to have succeeded in imposing some sort of order on the tangle — and

that the final product showed a family resemblance to the earlier

products. Finally, Robby was quick to jump to conclusions (his new

theory) in explaining why some figures were connected and others were

not.

Post Script — 4/3/78

After writing the preceding, I spoke to Robby again of his

game and his theory, inquiring whether or not he could prove it correct.

His method of choice was to test the case of 7 circles (which, as he

later found, splits into two planar figures of overlapping near-squares).

I tried to introduce the idea of a proof in place of another case study,

suggesting he take all possible cuttings of 3 connected circles and

figure out which one cuts the strip-figure in half. He said he had cut

the center first one time and at another had cut from one end.

Robby then drew the two pictures below on my chalk board:

He argued that it is always the last cut that severs the strip-figure

in two, representing the situation as at the above left. By cutting

along the dotted line, one joins the two small circles (here he made

motions of pulling apart the strip pieces) into the one large one.

Note well that this argument is merely a restatement of how he

appreciates the deformation, but it contented him.