### Vn77.1 **A Geometric Puzzle** 8/29 & 31/77

8/29 Since Miriam’s completion of our work with picture puzzles (cf.

Logo Session 40, 8/1/77), it has been my intention to examine her

performance with geometric puzzles. In the past, she has played with a

puzzle, the Pythagorean puzzle, which I had made from wood.

The Pythagorean puzzle is of 5 pieces and fits together in two ways.

The first fitting provides a square whose sides are the hypotenuses of

4 congruent triangles. The second fitting may be seen as the contiguous

placement of two smaller squares whose sides are the same length as the

two sides of the four congruent triangles.

Today I found Miriam and Robby playing on the floor of Glenn Iba’s

office with a small geometric puzzle. Robby played with a plastic

version and Miriam with a duplicate cut from cardboard. The pieces

below form squares in two ways:

Pieces 1 through 4 fit together to form a square. Pieces 1 through 5

form a slightly larger square.

During their play in Glenn’s office, Robby accepted a proffered

hint (Glenn first showed him the outline of the square and the location

of one piece). Miriam first refused to look at the hint Glenn offered,

then got mad at him when later he refused to show it to her. I brought

the cardboard puzzle home and put it on my desk for later use.

8/31 I found Miriam working at the puzzle this morning. She succeeded

relatively rapidly at the 4-piece assembly. As Robby tried to show her

Glenn’s hint, the arrival of the mailman drew the children away from

that task. Gretchen picked up the pieces, assembled the four, and left

it on a chair near the reading alcove.

Later Miriam joined me and tackled the 5-piece assembly. She failed.

She went over to my book shelves and took out the Pythagorean puzzle as

she said, “I’m going to give myself a good hint.” Miriam successfully

assembled the Pythagorean puzzle in both forms, but did not find that

success useful with Glenn’s puzzle. She decided first to make a design,

then asked for my help.

I had seen Glenn’s hint. I recalled the orientation of piece 1

with respect to the square’s edge and showed it to her. I noted that

the edge with 5 pieces was bigger than the edge with 4 pieces and set

as a sub-task finding a combination with edge length equal to that of

piece 1 with piece 5 inset. Once we found the place of piece 2, thus,

success was in reach. Miriam attempted piece 3 and failed repeatedly.

I recalled to her mind the picture puzzle hint: rotate the pieces.

Miriam then fit pieces 3 and 4 in place. Miriam is very happy and says,

“Robby thinks he’s the only one who can do this.” Miriam shows Gretchen

she can assemble the puzzle, then calls Robby to witness her success.

Before lunch, Miriam encountered the puzzle disassembled on the

dining room table. She talked to herself as she tried to assemble the

5-piece variation: “I’ve got a forgetting bug about this puzzle. . . . That

can’t be right. . . yep.” Miriam gives up and gets a snack.

In the afternoon, Miriam retries Glenn’s puzzle. She clearly

remembers the relation and placement of pieces 2 and 3. She also states

explicitly that piece 5 must be inset at the corner in piece 1, yet she

can not see how to fit the pieces together as she tries to place the 4th

piece adjacent to pieces 2 and 3. She is about to quit when I advise

her to rotate piece 4 once, then again, arriving at this arrangement:

at which point she sees how to fit the 1 – 5 combination into the 2-3-4

sub-assembly.

**Relevance**

Miriam’s puzzle assembly skill does not seem to generalize easily

from picture to geometric puzzles, nor from one geometric puzzle to

another. She knows when she is frustrated that she needs a ‘good hint’

and can apply it when given specific advice (note, however, that she

had to be directed to rotate piece 4 two times; she interpreted the

hint as: turn piece 4 so the next edge is adjacent to the 2-3 assembly,

instead of turn piece 4 until the configuration can accommodate sub-

assembly 1-5).