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CBSE · Class 8 · Computational Thinking and AI

A Square and a Cube: Thinking Puzzles

Standard difficulty · 20 questions · 40 marks · Time: 36 minutes

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
A perfect square can never end in which digit?
  1. 4
  2. 9
  3. 1
  4. 7
2.
50 × k is a perfect cube. What is the smallest natural number k?
  1. 40
  2. 2
  3. 50
  4. 20
3.
What is the smallest natural number k such that 1,400 ÷ k is a perfect square?
  1. 16
  2. 28
  3. 14
  4. 100
4.
How many perfect squares lie between 138 and 554?
  1. 14
  2. 12
  3. 13
  4. 11
5.
157,464 is a perfect cube. What is its cube root?
  1. 59
  2. 64
  3. 44
  4. 54
6.
A number machine takes the input, squares it, adds 20, then multiplies by 4. What comes out when 6 goes in?
  1. 224
  2. 228
  3. 56
  4. 596
7.
1 = 12, 1 + 3 = 22, 1 + 3 + 5 = 32. Following the pattern, what is 1 + 3 + 5 + … + 21?
  1. 121
  2. 144
  3. 100
  4. 132
8.
How many whole numbers lie strictly between 152 and 162?
  1. 30
  2. 31
  3. 29
  4. 15
Section B — Short answer8 × 2 = 16 marks
9.
Why can 2,000 not be a perfect square? Use its last digits.
10.
Find the cube root of 29,791 using prime factorisation.
11.
Find √6,241.
12.
What is the smallest whole number that must be added to 1,109 to make a perfect square?
13.
How many perfect cubes are there from 1 to 6,910?
14.
13 = 12, 13 + 23 = 32, 13 + 23 + 33 = 62. Use the pattern to find 13 + 23 + … + 123.
15.
Which of these are perfect squares? Write them in the order given: 606, 576, 16, 149, 246, 400
16.
How many numbers from 1 to 30,874 are both perfect squares and perfect cubes?
Section C — Word problems4 × 4 = 16 marks
17.
A gardener has 195 square tiles and wants to lay the largest possible square with them. (a) How many tiles long is each side? (b) How many tiles are left over? (c) How many more tiles would she need for a square one tile longer on each side?
18.
A cube of side 10 cm is built from 1,000 unit cubes, and its outside is painted. How many unit cubes have (a) three painted faces (b) exactly two (c) exactly one (d) none?
19.
For n = 300: (a) find the smallest k for which n × k is a perfect square (b) the square root of that square (c) the smallest m for which n × m is a perfect cube (d) the cube root of that cube.
20.
3,842 lies between two consecutive perfect squares. (a) What is the smaller one? (b) the larger one? (c) How many whole numbers lie strictly between them? (d) How many of those are less than 3,842?
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CBSE · Class 8 · Computational Thinking and AI

Answer key

A Square and a Cube: Thinking Puzzles · AW-8CT-SQC-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(d) 7squares end in 0, 1, 4, 5, 6 or 9
2(d) 2050 = 2 × 52; raise every exponent to a multiple of 3: k = 20, and 50 × 20 = 1,000 = 103
3(c) 141,400 = 23 × 52 × 7; remove one of each prime with an odd exponent: k = 14, and 1,400 ÷ 14 = 100 = 102
4(b) 12144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529
5(d) 54units digit 4 → cube root ends in 4; 157 lies between 53 and 63 → tens digit 5
6(a) 2246 → 36 → 56 → 224
7(a) 12121 is the 11th odd number, so the sum is 112 = 121
8(a) 30256 − 225 − 1 = 30
9A perfect square ends in an even number of zeros; 2,000 ends in three zeros.—
103129,791 = 313; take one prime from each group of three: 31 = 31
11796,241 = 792; halve each exponent: 79
1247332 = 1,089 < 1,109 < 1,156 = 342; 1,156 − 1,109
1319193 = 6,859 ≤ 6,910 < 8,000
146,0841 + 2 + … + 12 = 78; 782 = 6,084
15576, 16, 400576 = 242; 16 = 42; 400 = 202
165such numbers are sixth powers: 1, 64, 729, 4,096, 15,625
17(a) 13; (b) 26; (c) 1132 = 169 ≤ 195 < 196
18(a) 8; (b) 96; (c) 384; (d) 512corners 8; edges 12 × 8; faces 6 × 82; inside 83; check: 1000 = 103
19(a) 3; (b) 30; (c) 90; (d) 30300 = 22 × 3 × 52
20(a) 3,721; (b) 3,844; (c) 122; (d) 120612 and 622; between them 2 × 61 numbers