© Ivana Martić
Author: Nikola Sadovek
Challenge
In Santa Claus’s wrapping room, the elves are rushing to finish wrapping the last presents in time for Christmas Eve.
But there is a problem – the wrapping-paper machine is broken!
Instead of cutting all possible shapes of wrapping paper, the machine can only cut the paper into five special shapes (see Figure 1).
Each shape consists of so-called identical unit elf squares (all small squares have exactly the same size), and the machine can neither rotate nor flip them – every piece must be used exactly as it is shown in Figure 1.
Figure 1: The five special shapes.
The elves now need to cut some rectangular pieces of size m \times n with the machine, where m,n \geq 1. Here, m is the number of rows and n is the number of columns of the rectangular piece.
They want to determine for which pair (m,n), listed in the possible answers below, the entire rectangle can be filled using only these five shapes, with no leftover space. Can you help them figure it out before Santa leaves for his midnight journey?
Possible Answers
- m = 2021,\; n = 2024
- m = 2021,\; n = 2029
- m = 2022,\; n = 2026
- m = 2022,\; n = 2028
- m = 2023,\; n = 2024
- m = 2023,\; n = 2027
- m = 2024,\; n = 2026
- m = 2024,\; n = 2027
- m = 2024,\; n = 2029
- None of the other answers are true.