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NCERT Solutions Ganita Prakash (Part 1) Chapter 2 Chapter Challenge — Expression Engineer!

Book page 45 Updated on2026-09-05

Q1.
Using four 4’s, create expressions to get all values from 1 to 20.
Answer

Use exactly four 4s each time, with +, –, ×, ÷, brackets and the two-digit number 44.

ValueExpression
144 ÷ 44
24 ÷ 4 + 4 ÷ 4
3(4 + 4 + 4) ÷ 4
44 + 4 × (4 – 4)
5(4 × 4 + 4) ÷ 4
64 + (4 + 4) ÷ 4
744 ÷ 4 – 4
84 + 4 + 4 – 4
94 + 4 + 4 ÷ 4
10(44 – 4) ÷ 4
12(44 + 4) ÷ 4
154 × 4 – 4 ÷ 4
164 + 4 + 4 + 4
174 × 4 + 4 ÷ 4
20(4 + 4 ÷ 4) × 4

With only +, –, × and ÷ (and the two-digit 44), the values 11, 13, 14, 18 and 19 cannot be reached with exactly four 4s. They become possible once a decimal point is allowed:

11 = 4 ÷ .4 + 4 ÷ 4
13 = 4 ÷ .4 + 4 – 4 ÷ 4  (needs a fifth 4 — see the note)
14 = 4 + 4 + 4 + 4 ÷ .4  (also a five-4 answer)
18 = 4 ÷ .4 + 4 + 4
19 = 4 + 4 ÷ .4 + 4 + ...

The honest answer is this: 11, 13, 14, 18 and 19 have no four-4 expression using only +, –, × and ÷. Puzzle books allow the square root and the factorial for these, for example 19 = 4! – 4 – 4 ÷ 4 and 14 = 4 × 4 – 4 ÷ √4.

Why it happens: Four 4s and four operations reach only a limited set of numbers, because every 4 must be used once and the results of +, –, × and ÷ on 4s land on a coarse grid. Puzzle-makers therefore allow extra tools — the decimal point (.4), the square root and the factorial — to complete the run from 1 to 20.
Try This: How many of 1 to 20 can you reach with four 4s if 44 and 444 are allowed but nothing else? Compare your list with a friend’s.
Q2.
Using the numbers 1, 2, 3, 4, and 5 exactly once in any order get as many values as possible between –10 and +10.
Answer

Use each of 1, 2, 3, 4, 5 once, with +, –, ×, ÷ and brackets. Every whole number from –10 to 10 can be reached.

ValueExpressionValueExpression
105 + 4 + 3 – 2 × 1–11 + 2 – 3 + 4 – 5
91 + 2 – 3 + 4 + 5–22 + 4 – 3 – 5 × 1
84 + 5 + 2 – 3 × 1–31 + 2 + 3 – 4 – 5
71 + 2 + 3 – 4 + 5–41 × 2 + 3 – 4 – 5
62 × 3 + 4 – 5 + 1–51 – 2 – 3 + 4 – 5
51 + 2 + 3 + 4 – 5–6(4 – 5) × (1 + 2 + 3)
41 × 2 + 3 + 4 – 5–71 – 2 + 3 – 4 – 5
31 – 2 + 3 – 4 + 5–8(4 – 5) × (1 + 3) × 2
25 × 2 – 4 × (3 – 1)–91 + 2 – 3 – 4 – 5
11 + 2 – 3 – 4 + 5–102 – 3 – 4 – 5 × 1
05 – 4 – 3 + 2 × 1
Why it happens: Using only + and – on 1, 2, 3, 4, 5 always gives an odd total, because 1 + 2 + 3 + 4 + 5 = 15 is odd and flipping a sign changes the sum by an even amount. To land on an even value we must bring in × or ÷ — that is why the even rows above use multiplication.
Check it yourself: 5 × 2 – 4 × (3 – 1) = 10 – 8 = 2 ✓ and (4 – 5) × (1 + 3) × 2 = (–1) × 4 × 2 = –8 ✓
Q3.
Using the numbers 0 to 9 exactly once in any order, make an expression with a value 100.
Answer

Here is one neat solution using each of 0, 1, 2, …, 9 exactly once.

0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 × 9

Terms: 0, 1, 2, 3, 4, 5, 6, 7 and 8 × 9
= (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7) + 72
= 28 + 72
= 100

Two more, if two-digit numbers are allowed:

80 + 19 + 3 + 4 – 5 – 6 + 7 – 2 + 0 = 100
91 + 5 + 8 + 4 – 7 – 6 + 3 + 2 + 0 = 100
Why it happens: The trick is to make one big term that carries most of the value. Here 8 × 9 = 72 does the heavy lifting, and the remaining digits 0 to 7 add up to exactly 28 — the amount still needed to reach 100.
Check it yourself: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 and 8 × 9 = 72, so 28 + 72 = 100 ✓ Every digit from 0 to 9 has been used once.
Q4.
What other similar interesting questions can you ask?
Answer

Here are some puzzles of the same family that you can set for your class.

  • Using three 5s and any operations, make every value from 1 to 10.
  • Using four 2s exactly once each, which numbers from 1 to 15 can you make?
  • Using the digits of the current year (for example 2, 0, 2 and 6) once each, make the numbers 1 to 12 — one for each month.
  • Using 1, 2, 3, 4 exactly once each, what is the largest value you can make? And the smallest?
  • Place brackets in 8 – 3 × 2 + 4 to get as many different values as possible.
  • Using only the digit 7 and the four operations, write an expression whose value is 100.
  • Which numbers from 1 to 30 can be written as a sum of two or more consecutive whole numbers?
Why it happens: Every puzzle of this kind asks the same real question — how many different values can a fixed set of numbers produce once we are free to choose the operations, the order and the brackets. That is exactly what this chapter is about.
Tip: When you invent a puzzle, always solve it yourself first and note down at least one answer, so you know it is possible.
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