NCERT Solutions Ganita Prakash (Part 1) Chapter 4 –101Matchstick Patterns — In-text Questions

Book page 100 Updated on2026-09-05

Q1.

Look at the picture below. It is a pattern using matchsticks. Can you identify what the pattern is?

Answer

Each step adds one more triangle to the row, using only 2 extra matchsticks each time, because the new triangle shares one side with the previous one.

Step number	1	2	3	4	5	6
Triangles	1	2	3	4	5	6
No. of matchsticks	3	5	7	9	11	13

Why it happens: The first triangle needs all 3 sides. After that, the slanting side of the previous triangle is reused, so only 2 new sticks are needed.

Q2.

Can you tell how many matchsticks there will be in the next step, Step 5?

Answer

Step 4 uses 9 matchsticks, and every step adds 2.

Step 5 = 9 + 2 = 11 matchsticks

Q3.

How many matchsticks will there be in Step 33, Step 84, and Step 108? Of course, we can draw and count, but is there a quicker way to find the answers using the pattern present here?

Answer

Yes — count how many 2s are added. Step 1 starts with 3, and each later step adds one more 2.

Step 33 = 3 + 2 × 32 = 3 + 64 = 67
Step 84 = 3 + 2 × 83 = 3 + 166 = 169
Step 108 = 3 + 2 × 107 = 3 + 214 = 217

Tip: Drawing 108 triangles would take a whole page. The pattern gives the answer in one line.

Q4.

The number of matchsticks needed to make 33 triangles (Step 33) is _________. Similarly, find the number of matchsticks needed for Step 84 and Step 108.

Answer

Step 33 needs 67 matchsticks.

Step	Number of 2s added	Working	Matchsticks
33	32	3 + 2 + 2 + … (32 twos)	67
84	83	3 + 2 × 83	169
108	107	3 + 2 × 107	217

Why it happens: Step 2 has one 2 added, Step 3 has two 2s, Step 4 has three 2s. So Step y has (y – 1) twos added to the starting 3.

Q5.

What could be an expression describing the rule/formula to find out the number of matchsticks at any step? For step y, what is the expression?

Answer

Step y has one 3 and (y – 1) twos.

Number of matchsticks in Step y = 3 + 2 × (y – 1)

Noticing that the first step also contains a 2 (since 3 = 1 + 2) gives a second form:

2y + 1

Q6.

Does the above expression also give the number of matchsticks at each step correctly? Are these expressions the same?

Answer

Yes to both. Simplify the first expression and it turns into the second.

3 + 2 × (y – 1) = 3 + 2y – 2
= 2y + (3 – 2)
= 2y + 1

y	3 + 2(y – 1)	2y + 1
1	3	3
5	11	11
33	67	67

Why it happens: Two people can count the same picture in different ways and still be right. Their expressions look different but simplify to the same thing.

Q7.

Matchsticks are placed in two orientations — (a) horizontal ones at the top and bottom, and (b) the ones placed diagonally in the middle. For example, in step 2 there are 2 matchsticks placed horizontally and 3 matchsticks placed diagonally. What are these numbers in Step 3 and Step 4?

Answer

Step	Horizontal	Diagonal	Total
1	1	2	3
2	2	3	5
3	3	4	7
4	4	5	9

Step 3: 3 horizontal and 4 diagonal (3 + 4 = 7 ✔)
Step 4: 4 horizontal and 5 diagonal (4 + 5 = 9 ✔)

Q8.

How does the number of matchsticks change in each orientation as the steps increase? Write an expression for the number of matchsticks at Step ‘y’ in each orientation. Do the two expressions add up to 2y + 1?

Answer

Horizontal sticks go 1, 2, 3, 4, … and diagonal sticks go 2, 3, 4, 5, … Each set grows by exactly 1 per step.

Horizontal at Step y = y
Diagonal at Step y = y + 1

Sum = y + (y + 1) = 2y + 1 ✔

Why it happens: Each new triangle adds one horizontal stick and one slanting stick — that is the “2 more each time”. The extra diagonal comes from the very first triangle, which has two slanting sides.

Check it yourself: Step 6 → 6 horizontal + 7 diagonal = 13, and 2 × 6 + 1 = 13 ✔

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