In January, I paid a visit to MathsWorld, the recently opened maths discovery centre in London, alongside some other members of the Chalkdust team. One of the highlights of the trip was playing the two-player game Genius Square.
In Genius Square, you start with a six-by-six board and roll seven dice. These dice tell you where to place seven cylindrical blocks, for example:
The dice
The board with the cylinders placed
The two players then race to fit the pieces shown above into the remaining space on the board. The pieces that the players have are the five tetrominoes, the two triominoes, a domino, and a single square (or monomino); these are all the shapes you can make by gluing together up to four squares (if rotations and reflections are considered the same shape).
The pieces
There’s some clever design in this game: if, instead of rolling the dice, you were to randomly pick any set of seven spaces to place the cylinders, the puzzle is not guaranteed to have a solution. The locations printed on the dice have been carefully chosen so that any combination that you can roll leads to a solvable puzzle.
A puzzle-a-day
The Genius Square puzzle is similar to another rearrangement puzzle: the puzzle-a-day calendar, created by the Norwegian puzzle makers DragonFjord. In this puzzle, you are given the pieces below and asked to place them on the board to cover everything except today’s date.
The puzzle-a-day board
The puzzle-a-day pieces
For example, on 22 July, you could place the pieces as shown below.
A solution of puzzle-a-day for 22 July
DragonFjord make and sell wooden and plastic version of puzzle-a-day, which you can buy from Maths Gear—who also provide the top prize for the crossnumber (see the Issue 23 crossnumber)—to avoid the cost of shipping directly from Norway.
In puzzle-a-day, it’s possible to arrange the pieces to make every single combination of a number and a month, including days that don’t exist like 31 September and 30 February. While we were considering options for the cover of this issue, we discussed putting something like Genius Square on the cover, and I began to wonder if it would be possible to make a puzzle like puzzle-a-day but where it was only possible to make days that actually appear on the calendar.
A solution of puzzle-a-day for 31 September?!
A new puzzle
After spending a while scribbling on squared paper and getting nowhere, I had an idea: I could put the months in regions that were disconnected from the day numbers. Then, by carefully choosing the shape of the month regions and the arrangement of the dates, I could force the solver to use different combinations of pieces on the day numbers for different months.
Once I’d had this idea, I threw together some Python code that could see which day numbers you could and couldn’t leave uncovered with a set of pieces, and waited for it to find a good set of pieces. It found the board and pieces shown to the right. As in Tetris, I’ve named the pieces after letters that they vaguely resemble.
In January, March, May, July, August, October and December, you have to use a P, an O and the A in the month regions. The remaining pieces can make any day from 1 to 31.
In April, June, September and November, you need to use the C, an O and the A in the month regions. This leaves pieces that can make any day from 1 to 30, but importantly can’t make 31.
In February, you need to use both Os and the C in the month regions. This leaves pieces that can make any day from 1 to 29, but not 30 or 31.
Now all we need to do is find another new arrangement that somehow works differently in leap and non-leap years…
The board for the new puzzle
The pieces for the new puzzle
More from Chalkdust
In conversation with Anne Skeldon
Sam Kay and Ellen Jolley race against the body clock with the University of Surrey professor
Cheating at Tetris
Ffreuer Bristow works out which pieces can force someone to lose
The secret life of words
Barry Pawlik builds words and sentences from doubles
Solving a cubic in 1646
Elinor Flavell takes squaring the circle to another level
Coffee problems
Patrick Hedfeld has bean there, modelled that
The beautiful soaisu partition
Kenichi Takemura uncovers symmetries hidden in plain sight