Texas Mathematics Teacher Spring/Summer 2018 - 27

Pythagoras Unchained

Likewise, participants drew the segments in the second column (those with an
x-coordinate of 2) (see Figure 3).
Figure 3. Drawn segments with an x-coordinate of 2.
Again, using the Pythagorean theorem, participants started at the bottom and
calculated the lengths of these segments. They were: 4, 5, 8, 13, 20, 29 .
Continuing for all five columns participants produced a table of segment lengths (see Figure 4).
Figure 4. Participants' exploration results for segment lengths using the Pythagorean theorem.
Column

Lengths

1

1

2

5

10

17

26

2

4

5

8

13

20

29

3

9

10

13

18

25

34

4

16

17

20

25

32

41

5

25

26

29

34

41

50

While Figure 4 reflects an accurate solution, the
authors saw some additional patterns. Because
this presentation of the problem has essentially
"unchained" Pythagoras from the usual context of
solving sides of a specific triangle, we thought there
might be a more compact and general description,
perhaps even a function which would generate all the
possible segment lengths. Therefore, we generalized
the problem to include all possible segment lengths.

The Problem and Investigation
Given an infinitely large piece of paper on which dots have been placed in a square grid pattern 1 unit apart,
what segment lengths can be produced?
To begin, we looked for a pattern in the table represented in Figure 4. For simplicity, we focused on the
radicands (see Figure 5). We will call the list of radicands from column 1, sequence 1. We will call the list of
radicands from column 2, sequence 2, and so on.
Figure 5. The radicands from the participants' exploration using the Pythagorean theorem.
Column
Radicands
1
1 2 5 10 17
2
4 5 8 13 20
3
9 10 13 18 25
4
16 17 20 25 32
5
25 26 29 34 41

26
29
34
41
50

Pythagoras of Samos
c. 570 - c. 495 BC
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