The Matchstick Growth Pattern
A learner builds a sequence of shapes from matchsticks. Shape 1 is a single square made of 4 matchsticks. Shape 2 is two squares side by side sharing one side, using 7 matchsticks. Shape 3 is three squares side by side, using 10 matchsticks. The pattern continues, each new square added to the right sharing one side with the previous square. The learner wants a rule for the number of matchsticks in shape n, and wants to know which shape uses exactly 100 matchsticks.
1. Find a rule (formula) for the number of matchsticks in shape n, and verify it for the first three shapes.
Read the worked answer
Look at the numbers: shape 1 uses 4, shape 2 uses 7, shape 3 uses 10. Each new shape adds 3 matchsticks (one new square needs 4 sticks but shares 1 side, so adds 3).
This is a linear pattern with common difference 3. The rule is: matchsticks = 3n + 1.
Check: n = 1 → 3(1) + 1 = 4 ✓; n = 2 → 3(2) + 1 = 7 ✓; n = 3 → 3(3) + 1 = 10 ✓.
Reasoning: the first square uses 4 sticks; each additional square adds 3, giving 4 + 3(n − 1) = 3n + 1.
2. Which shape number uses exactly 100 matchsticks? Solve and explain whether such a shape exists.
Read the worked answer
Set 3n + 1 = 100. Then 3n = 99, so n = 33.
Check: 3(33) + 1 = 99 + 1 = 100 ✓.
So shape 33 uses exactly 100 matchsticks. Since 33 is a whole number, such a shape exists.
3. A classmate claims that some shape will use exactly 101 matchsticks. Use the rule to judge this claim.
Read the worked answer
Set 3n + 1 = 101. Then 3n = 100, so n = 100 ÷ 3 = 33⅓.
Since n must be a whole number (you cannot have a third of a shape), no shape uses exactly 101 matchsticks.
We can also see this from divisibility: 3n + 1 = 101 requires 100 to be divisible by 3, but 1 + 0 + 0 = 1, which is not a multiple of 3. So the claim is false. The matchstick totals are 4, 7, 10, 13, …, always 1 more than a multiple of 3; 101 is not of this form (101 − 1 = 100 is not a multiple of 3), so it never appears.