Problem 20 · 2014 Math Kangaroo
Stretch
Counting & Probability
sequence-of-figurescareful-counting
The black diamonds ◆ and white diamonds ◇ follow a fixed pattern. The first 3 levels are shown on the right. Each level (from the 2nd level on) has one more row than the level before. In every level, the two outermost diamonds of the last row are white, and all the other diamonds are black. How many black diamonds are there in level 6?
Show answer
Answer: C — 26
Show hints
Hint 1 of 3
Each level is a little triangle of rows: 1 diamond, then 2, then 3, and so on down.
Still stuck? Show hint 2 →
Hint 2 of 3
Count ALL the diamonds in level 6 first, then take away the white ones.
Still stuck? Show hint 3 →
Hint 3 of 3
In every level only the two ends of the very bottom row are white.
Show solution
Approach: count all the diamonds, then take away the two white ones
- Level 6 has 6 + 1 = 7 rows, with 1, 2, 3, 4, 5, 6, 7 diamonds in them.
- Adding those gives 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 diamonds altogether.
- Only the two ends of the bottom row are white, so the black ones are 28 − 2 = 26.
- Answer: 26.
Mark:
· log in to save