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NCERT Solutions Ganita Prakash (Part 1) Chapter 1 Puzzle Time — Puzzle Time: Toothpick Digits

Book page 23 Updated on2026-09-05

Q1.
We can write digits using sticks as shown in the image. To make the digit 7, three sticks are needed. Write or make the number 5108. How many sticks are required?
Answer

20 sticks.

Digit0123456789
Sticks6255456376
5 → 5 sticks
1 → 2 sticks
0 → 6 sticks
8 → 7 sticks
Total = 5 + 2 + 6 + 7 = 20 sticks
Why it happens: Each digit is drawn on the same seven-segment frame, so the stick count depends only on how many of the seven segments are lit. 8 uses all 7 and is the costliest; 1 uses only 2 and is the cheapest. Keep this table handy — every question below uses it.
Q2.
1. Make or write the number 42,019. It would require exactly 23 sticks. 2. Starting with 42,019, add or write two more sticks, and make a bigger number. One example is 42,078. What other numbers bigger than 42,019 can you make in this way? 3. Preetham wants to insert the digit ‘1’ somewhere among the digits ‘4’, ‘2’, ‘0’, ‘1’ and ‘9’. Where should he place the digit ‘1’ to get the biggest possible number? 4. What other numbers can he make by placing the digit ‘1’?
Answer

1. 42,019 needs 4 + 5 + 6 + 2 + 6 = 23 sticks ✓

2. Two extra sticks means the new number must use exactly 25 sticks and be bigger than 42,019. Some examples:

42,022 (4+5+6+5+5 = 25)  42,033  42,053  42,064
42,078 (4+5+6+3+7 = 25) — the book's example
42,087  50,019  50,073  99,974 (6+6+6+3+4 = 25) — the largest possible

3. He should place the ‘1’ between the ‘2’ and the ‘0’, giving 4,21,019.

1 42019 → 1,42,019
4 1 2019 → 4,12,019
4 2 1 019 → 4,21,019 ← biggest
4 2 0 1 19 → 4,20,119
4 2 0 1 9 1 → 4,20,191

4. The other numbers he can make are 1,42,019 · 4,12,019 · 4,20,119 · 4,20,191.

Why it happens: Inserting the 1 pushes every digit to its right one place further left, so you want to delay that push until after the biggest digits. Putting the 1 right after ‘42’ keeps 4 and 2 in the two highest places while lifting the whole number to six digits. Note that slipping the 1 just before or just after the existing 1 gives the same number, 4,20,119 — so there are only five different results, not six.
Q3.
1. Make or write the number 63,890. 2. Starting with 63,890, rearrange exactly four sticks and make a bigger number. One example is 88,078. What other numbers bigger than 63,890 can you make in this way?
Answer

1. 63,890 needs 6 + 5 + 7 + 6 + 6 = 30 sticks.

2. Rearranging (not adding or removing) sticks keeps the total at 30 sticks. So look for 5-digit numbers bigger than 63,890 that also use exactly 30 sticks:

88,078 = 7+7+6+3+7 = 30 ✓ (the book's example)
63,896 = 6+5+7+6+6 = 30 ✓
63,908 = 6+5+6+6+7 = 30 ✓
63,980 = 6+5+6+7+6 = 30 ✓
64,880 = 6+4+7+7+6 = 30 ✓
99,999 = 6+6+6+6+6 = 30 ✓ — the largest possible
Why it happens: Moving a stick from one place to another never changes the total number of sticks, so the target number must have the same stick count as the original — 30. Among all 5-digit numbers with a stick count of 30, the largest is 99,999, since 9 costs 6 sticks and 6 × 5 = 30 exactly.
Check it yourself: After listing candidates by stick count, look at the actual pictures to confirm that exactly four sticks (no more, no fewer) really do have to move. 63,890 → 63,896 needs the 0 turned into a 6 and the 9 into a 9 — try drawing it.
Q4.
1. Make any number using exactly 24 sticks or lines. 2. What is the biggest number that can be made using 24 sticks or lines? 3. What is the smallest number that can be made using 24 sticks or lines?
Answer

1. Many numbers use exactly 24 sticks. Three easy ones:

8887 → 7 + 7 + 7 + 3 = 24 ✓
4,44,444 → 4 + 4 + 4 + 4 + 4 + 4 = 24 ✓
2008 → 5 + 6 + 6 + 7 = 24 ✓

2. The biggest number is 111111111111 — twelve 1's.

1 costs the fewest sticks, only 2.
24 ÷ 2 = 12 digits
Thirteen digits would need at least 13 × 2 = 26 sticks > 24
So the most digits possible is 12, and all of them must be 1's
Biggest number = 1,11,11,11,11,111 (one arab eleven crore …)

3. The smallest number is 2008.

Fewest digits: the costliest digit 8 uses 7 sticks,
so 3 digits give at most 21 sticks — not enough.
Hence at least 4 digits.
First digit 1 (2 sticks) leaves 22 for 3 digits, but 3 × 7 = 21 < 22 — impossible.
So the first digit is 2 (5 sticks), leaving 19 for three digits.
Next digit as small as possible: 0 (6 sticks), leaving 13 for two digits.
Next digit 0 again (6 sticks), leaving 7 → the last digit is 8.
Smallest = 2008  (5 + 6 + 6 + 7 = 24 ✓)
Why it happens: More digits always beat bigger digits when you want a large number, so the biggest answer uses the cheapest digit as many times as possible. For the smallest number the logic flips — you want the fewest digits, and then the smallest digit you can afford in each place from the left.
Try This: Make your own questions. What is the biggest number using 20 sticks? (Ten 1's: 1111111111.) What is the smallest using 20 sticks? (Try it — the answer has 3 digits.)
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