MathDB

5

Part of 2003 May Olympiad

Problems(2)

shortest path of movement of an ant on the surface of a cube, returning in start

Source: May Olympiad (Olimpiada de Mayo) 2003 L2

9/15/2018
An ant, which is on an edge of a cube of side 88, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.
geometry3D geometrycubeminimum
chess knight moves on 2x2 board

Source: IX May Olympiad (Olimpiada de Mayo) 2003 L1 P5

9/22/2022
We have a 4×44 \times 4 squared board. We define the separation between two squares as the least number of moves that a chess knight must take to go from one square to the other (using moves of the knight). Three boxes A,B,CA, B, C form a good trio if the three separations between AA and BB, between AA and CC and between BB and CC are equal. Determines the number of good trios that are formed on the board.
Clarification: In each move the knight moves 22 squares in the horizontal direction plus one square in the vertical direction or moves 22 squares in the vertical direction plus one square in the horizontal direction.
combinatorics