Finding the Unknown | FIO

Question 24

Two separate figures are given below. Each figure shows the first few positions in a sequence of arrangements made with sticks. Identify the pattern and answer the following questions for each figure:

(a) How many squares are in position number 11 of the sequence? (b) How many sticks are needed to make the arrangement in position number 11 of the sequence? (c) Can an arrangement in this sequence be made using exactly 85 sticks? If yes, which position number will it correspond to? (d) Can an arrangement in this sequence be made using exactly 150 sticks? If yes, which position number will it correspond to?

Solution

We will identify the pattern for squares in the first figure and sticks in the second figure.

Diagram 1

Step 1 — Squares in the first figure

Let $S_n$ be the number of squares in position $n$ of the first figure. We observe the pattern for the number of squares. Position 1 has $1 + 1 \times 3 = 4$ squares. Position 2 has $1 + 2 \times 3 = 7$ squares. Position 3 has $1 + 3 \times 3 = 10$ squares. The number of squares in position $n$ is $1 + 3n$ . We want to find the number of squares in position 11. We substitute $n = 11$ into the pattern.

$S_{11} = 1 + 3 \times 11$

$= 1 + 33$

$\boxed{34 \text{ squares}}$

Diagram 2

Step 2 — Sticks in the second figure for position 11

Let $T_n$ be the number of sticks in position $n$ of the second figure. We observe the pattern for the number of sticks. Position 1 needs $13 + 9 \times (1-1) = 13$ sticks. Position 2 needs $13 + 9 \times (2-1) = 22$ sticks. Position 3 needs $13 + 9 \times (3-1) = 31$ sticks. The number of sticks in position $n$ is $13 + 9(n-1)$ . We want to find the number of sticks in position 11. We substitute $n = 11$ into the pattern.

$T_{11} = 13 + 9 \times (11-1)$

$= 13 + 9 \times 10$

$= 13 + 90$

$\boxed{103 \text{ sticks}}$

Step 3 — Can 85 sticks be used?

We want to know if an arrangement can be made with 85 sticks. Let $n$ be the position number. We set the number of sticks $T_n$ equal to 85.

$13 + 9(n-1) = 85$

We subtract 13 from both sides.

$9(n-1) = 85 - 13$

$9(n-1) = 72$

We divide both sides by 9.

$n-1 = \frac{72}{9}$

$n-1 = 8$

We add 1 to both sides.

$n = 8 + 1$

$\boxed{n = 9}$

Since $n$ is a whole number, an arrangement can be made. It is the 9th arrangement.

Step 4 — Can 150 sticks be used?

We want to know if an arrangement can be made with 150 sticks. Let $n$ be the position number. We set the number of sticks $T_n$ equal to 150.

$13 + 9(n-1) = 150$

We subtract 13 from both sides.

$9(n-1) = 150 - 13$

$9(n-1) = 137$

We divide both sides by 9.

$n-1 = \frac{137}{9}$

$n-1 = 15.222...$

We add 1 to both sides.

$n = 15.222... + 1$

$\boxed{n = 16.222...}$

Since $n$ is not a whole number, no arrangement can be made with exactly 150 sticks.

Answer

(a) The number of squares in the 11th position of the first figure is 34. (b) The number of sticks needed for the 11th position of the second figure is 103. (c) Yes, an arrangement can be made using exactly 85 sticks. It corresponds to position number 9. (d) No, an arrangement cannot be made using exactly 150 sticks.

More questions in FIO

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Q19

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Q24

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