Coin Conjoin Puzzle

The Problem

You have a triangle of 10 coins (pointing up). You need to invert it (point down) by moving the minimum number of coins.

Solution Logic

1. Triangle T3 (6 coins):

   • Move the 3 corners? No.
   • Move just 2 coins.
   • Move the top coin to the bottom center.
   • Move two bottom corners up.

2. Triangle T4 (10 coins):

   • The triangle has 4 rows (1, 2, 3, 4).
   • To invert it, we basically need to remove the “corners” that define the shape and place them on the opposite side.
   • Minimum Moves: 3.
   • How:
     1. Move the top tip coin to the very bottom (below the row of 4).
     2. Move the two coins at the ends of the bottom row (row 4) to the top positions (forming the new base of the inverted triangle).

General Formula

For a triangle with side length $n$: $\text{Moves} = \lfloor \frac{n(n+1)}{6} \rfloor$

• $n=3$ (6 coins): $\lfloor 12/6 \rfloor = 2$.
• $n=4$ (10 coins): $\lfloor 20/6 \rfloor = 3$.
• $n=5$ (15 coins): $\lfloor 30/6 \rfloor = 5$.