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NCERT Solutions Ganita Prakash (Part 1) Chapter 6 Coin Conjoin — Puzzle Time: Coin Conjoin

Book page 158 Updated on2026-09-05

Q1.
Arrange 10 coins in a triangle as shown in the figure below on the left. The task is to turn the triangle upside down by moving one coin at a time. How many moves are needed? What is the minimum number of moves?
Answer

The 10-coin triangle has 4 rows: 1, 2, 3, 4 coins from the top. The upside-down triangle has 4, 3, 2, 1 from the top.

Lay the two triangles on top of each other: 7 coins already sit in the right places. Only 3 coins have to move.

  • Move the single apex coin down to below the bottom row.
  • Move the two end coins of the bottom row up beside the second row.
Minimum number of moves = 3

The book also records the smaller cases: a 3-coin triangle needs 1 move and a 6-coin triangle needs 2.

Why it happens: a coin need not move if it lies in a cell that belongs to both the upright and the inverted triangle. The largest such overlap for 10 coins is 7 cells, so at least 10 – 7 = 3 coins must move — and 3 is achievable, so 3 is the minimum.
Q2.
Find out the minimum possible moves needed to flip the next bigger triangle having 15 coins. Try the same for bigger triangular numbers.
Answer

The 15-coin triangle has 5 rows. Superimposing the two triangles leaves 10 coins in place, so 5 coins must move — and 5 moves are enough.

Rows nCoins T = n(n + 1)⁄2Minimum moves
231
362
4103
5155
6217
7289
15 coins: 5 moves
Q3.
Is there a simple way to calculate the minimum number of coin moves needed for any such triangular arrangement?
Answer

Yes. Compare the “coins” column with the “moves” column above: the number of moves is always one third of the number of coins, rounded down.

Coins in a triangle of n rows: T = n(n + 1)⁄2
Minimum moves = ⌊T ⁄ 3⌋ = ⌊n(n + 1) ⁄ 6⌋
nTT ⁄ 3Moves
3622 ✓
4103.33…3 ✓
51555 ✓
62177 ✓
8361212 ✓
Why it happens: a coin can stay put only if its cell lies in both triangles, and roughly two thirds of the cells can be made to overlap. So about one coin in three has to move — and the exact count is the whole-number part of T ⁄ 3.
Try This: check the rule on the 3-coin triangle: T = 3, T ⁄ 3 = 1, and indeed one move flips it. Then try to build the 21-coin flip in 7 moves and see whether you can do better — you will not be able to.
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