Reasoning · Grade 5-1 Rules and Correspondence

Problem

Count a growing figure three ways

Matchsticks make a strip of triangles. The first uses three, and each next one shares a match with the previous. The three expressions for 21 triangles are 1+2×21, 3+2×20 and 3×11+10, each giving 43. Explain how each of the three expressions counts them.
Your answer
How to solve
Strategy Look for a Pattern — First I build the strip from small cases and find the growth rule, so I know 21 triangles really do need 43 matches. Then, instead of reading the three expressions as arithmetic, I read them as three different ways of cutting the same picture into groups: an expression of the form (start) + (repeat) x (how many) says 'lay this down first, then add this much each time', and an expression of the form (size) x (count) + (extra) says 'sort the pieces into two kinds and count each kind'. Drawing the split for each expression is what turns a number sentence back into a story.
1STEP 1

Grow the strip and find the rule

Each extra triangle adds 2 matches.

3, 5, 7, 9, 11, … (+2 each time)
2STEP 2

Confirm the total for 21 triangles

So 21 triangles take 43.

2 × 21+1=43
3STEP 3

Method behind expression (1): 1 + 2 x 21

(1) lays one down, then 2 per triangle.

1 + 2 × 21 = 1 + 42 = 43
4STEP 4

Method behind expression (2): 3 + 2 x 20

(2) starts with 3, then 2 for each of the other 20.

3 + 2 × 20 = 3 + 40 = 43
5STEP 5

Method behind expression (3): 3 x 11 + 10

(3) gives 3 to each of 11 up-triangles and 1 to each of 10 down-triangles.

3 × 11 + 10 = 33 + 10 = 43
6STEP 6

Check that the up-and-down split really gives 11 and 10

The 11-and-10 split checks out, so all three give 43.

11 + 10 = 21, 11 - 10 = 1
Answer
43 matches
1 + 2 × 21 = 3 + 2 × 20 = 3 × 11 + 10 = 43
All three expressions equal 43: 1 + 42 = 43, 3 + 40 = 43, 33 + 10 = 43. That agrees with the growth rule 2 x n + 1 for n = 21, and with the picture, where n = 5 gives 2 x 5 + 1 = 11 matchsticks - exactly the 11 drawn. A sanity bound also fits: 21 separate triangles would need 63 matchsticks, and every join saves one, with 20 joins in the strip, so 63 - 20 = 43. Each description uses precisely the numbers that appear in its expression, and the count of triangles in method (3), 11 + 10, comes back to 21.
Takeaway

The same 43 matchsticks can be bundled three different ways - what changes is only where you decide the counting starts.

  • Grow the strip and find the rule
  • Confirm the total for 21 triangles
  • Method behind expression (1): 1 + 2 x 21
  • Method behind expression (2): 3 + 2 x 20
  • Method behind expression (3): 3 x 11 + 10
  • Check that the up-and-down split really gives 11 and 10