Evdokimov M.A.

Place 32 knights on a chessboard so that each of them attacks exactly two others.

#
In a $3 \times 3$ square, it is possible to visit all cells except the central one with a knight's move and return to the starting cell.

## Solution

If knights are placed on all cells of a $3 \times 3$ square except the central one, each knight will attack exactly two others. Now, place four such "squares" on the board far enough apart so that knights from different squares do not attack each other (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_b3d98ec29b07c8345931g-19.jpg?height=432&width=426&top_left_y=2288&top_left_x=816)

Author: Shapovalov A.V.

On the Moon, coins with denominations of 1, 15, and 50 ferthings are in circulation. Nезнайка paid for a purchase with several coins and received change - one more coin than he paid with. What is the smallest possible price of the purchase?

## Hint

The remainder when 1, 15, and 50 are divided by 7 is 1.

## Solution

The remainder when the denomination of each coin is divided by 7 is 1. Let Nезнайка pay with $k$ coins totaling $A$ and receive change $C$. Then $A \equiv k(\bmod 7)$, and

$C \equiv k+1(\bmod 7)$. Therefore, the price of the purchase $A-C \equiv 6(\bmod 7)$. Consequently, this price cannot be less than 6 ferthings.

An example where the purchase costs 6 ferthings: Nезнайка paid with two coins - 1 ferthing and 50 ferthings, and received change with three coins of 15 ferthings each.

## Answer

6 ferthings.