A pile of coins of the same size, round.
Question one.
Eight coins are lined up in a row, how many moves are needed to divide them into four piles (of two each)? The rules for moving are that the coins can only jump in the direction of the arrangement, they must cross two coins at a time, and two coins can overlap in the same position.
Solution:
Numbering the coins 1 through 8 requires four steps
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 5 | 6 | 47 | 8 | |
| 1 | 26 | 3 | 5 | 47 | 8 | ||
| 13 | 26 | 5 | 47 | 8 | |||
| 13 | 26 | 47 | 58 |
Question two: How many times does a rolling coin roll when n coins meet edge to edge to form a circle, and an extra coin is rolled around them? The edges must be touched during rolling.
Solution:
No matter what shape these coins are rolled around, the extra coin always makes the same number of revolutions. This is because the center of the coin’s circle always passes the same distance.
- 2 turns around 1 coin.
- 3 revolutions around 3 coins
- 5 revolutions around 9 coins
The sum of the interior angles of the polygons enclosed by the centers of the coins (concave polygons are included under the rule) is fixed plus 60° lost at each junction.
\[N=(360n-((n-2) \times 360 + 120 n))/360\times 2 =n/3+2\]