This document contains a series of puzzles involving building rectangles and squares out of smaller squares. It notes that an earlier posting of one of the puzzles mistakenly included a 5 in one square, which was impossible. The author thanks a student for catching this mistake, and acknowledges intentionally including impossible problems in classes to encourage proving them impossible, though this was an unintentional error.

1. 1
7
A rectangle built
from 9 squares

2. 1
31
A rectangle built
from 9 squares

3. 43
34
A rectangle or
square built from
11 squares.

4. 7
4
68
2
3
A rectangle or
square built from
22 squares.

5. 14
35 33
9
A rectangle or
square built from
24 squares.

6. 37
35
29
19
A square built of
the fewest different
squares.

7. 14
7
6
9
15
24

8. 14
7
5
9
15
24
I made a mistake
posting this puzzle the
first time. I wrongly
put a 5 in this square!
See that the two
squares with an X will
always have the same
size!
x
x
Thanks to Kaela Worrell’s
daughter for finding this! I
only half apologize for my
mistakes. Finding impossible
problems and proving that
they are impossible is a very
rich activity! In my own
classes I purposely do it, but
this, alas was an unintentional
mistake. Ouch ;-)