Question 5
In Silicon Valley, south of San Francisco Bay, Dr John invents three types of chips, A, B and C, and stores them in boxes with different locks encoded by square grids with some cells removed. A box opens if the remaining cells can be covered exactly by the illustrated polyominoes, with no overlap. Some locks were carelessly designed and cannot open. Investigate all the following questions.
1. Box A has the illustrated 6×6 board with two black cells removed. Can Dr John open it using 17 dominoes? If so, show a tiling.
2. Box B has a different illustrated 6×6 board with two cells removed. Dr John cannot tile it with 17 dominoes, so he colours the full board alternately green and yellow as shown. How many cells of each colour are there? What colours are the removed cells, and how many of each colour remain? How many green and yellow cells does each domino cover, and how many would 17 dominoes cover? By comparing these numbers, decide whether box B can open; show a tiling if it can.
3. Box C has the illustrated 6×6 board with three cells removed and requires 11 straight I-triominoes. Dr John observes that the preceding two-colour test does not settle it, and colours the board in the illustrated repeating three-colour pattern instead. Using this colouring, decide whether box C can open; show a tiling if it can.
1. Box A has the illustrated 6×6 board with two black cells removed. Can Dr John open it using 17 dominoes? If so, show a tiling.
2. Box B has a different illustrated 6×6 board with two cells removed. Dr John cannot tile it with 17 dominoes, so he colours the full board alternately green and yellow as shown. How many cells of each colour are there? What colours are the removed cells, and how many of each colour remain? How many green and yellow cells does each domino cover, and how many would 17 dominoes cover? By comparing these numbers, decide whether box B can open; show a tiling if it can.
3. Box C has the illustrated 6×6 board with three cells removed and requires 11 straight I-triominoes. Dr John observes that the preceding two-colour test does not settle it, and colours the board in the illustrated repeating three-colour pattern instead. Using this colouring, decide whether box C can open; show a tiling if it can.