Reasoning · Grade 4-1 Working with Sequences

Problem

Sierpinski triangle stage counting

Every black triangle is cut into 4 using the midpoints of its sides. The middle one is thrown away. Stage 0 has 1 black triangle, stage 1 has 3, stage 2 has 9. Draw stage 3 and count the black triangles in stage 4.
0th 1st 2nd …… Each stage divides every black triangle into 4 and removes the middle one.
Your answer
How to solve
Strategy Look for a Pattern — Drawing the 4th stage would mean shading 81 tiny triangles, which is unreasonable by hand. Instead I look at what one step of the rule does to a single black triangle, check that against the three printed stages, and turn it into a counting rule. Drawing the 3rd stage for part (1) is small enough to do and it also confirms the counting rule one more time before it is used for stage 4.
1STEP 1

Count the black triangles in the printed stages

The printed stages hold 1, 3 and 9 black triangles.

0th: 1, 1st: 3, 2nd: 9
2STEP 2

See what the rule does to one black triangle

One black triangle becomes 3 black ones.

4 - 1 = 3 black pieces from every 1 black piece
3STEP 3

Check the times-3 rule against every printed stage

The times-3 rule fits every printed stage.

1 × 3 = 3 ✓ 3 × 3 = 9 ✓
4STEP 4

Part (1): draw the 3rd Sierpinski triangle

Stage 3 gives 9 × 3 = 27.

9 × 3 = 27 black triangles in the 3rd stage
5STEP 5

Part (2): count the 4th stage

Stage 4 gives 27 × 3 = 81.

1 → 3 → 9 → 27 → 81
Answer
27, 81 black triangles
9 × 3 = 27, 27 × 3 = 81
The counts 1, 3, 9, 27, 81 grow but stay whole numbers, and each is 3 times the one before, which matches the fact that every black triangle becomes 3 black triangles. The answer should be bigger than the 27 of the 3rd stage and smaller than the number of pieces if nothing were removed: without any cutting away, stage 4 would have 4 x 4 x 4 x 4 = 256 small triangles, and 81 is comfortably less than that, as it must be since holes were removed at every stage. Counting a stage is a count of pieces, so a whole number with no units is the right kind of answer.
Takeaway

Every black triangle turns into 3 smaller black ones, so the count just keeps tripling: 1, 3, 9, 27, 81.

  • Count the black triangles in the printed stages
  • See what the rule does to one black triangle
  • Check the times-3 rule against every printed stage
  • Part (1): draw the 3rd Sierpinski triangle
  • Part (2): count the 4th stage