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A new puzzle every day
2026-04-20
This is an article that I wrote for
Chalkdust issue 23, and the
new puzzle it introduces appears on the cover of issue 23.
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:
![](javascript:showlimage("genius-square-dice.png"))
The dice
![](javascript:showlimage("genius-square-board.png"))
![](javascript:showlimage("genius-square-pieces.png"))
The board with the cylinders placed and the pieces
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).
If you want to ruin/improve your copy of Chalkdust, you could cut out the pieces shown above and
try to fit them in the board to the left.
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.
![](javascript:showlimage("puzzle-a-day-board.png"))
![](javascript:showlimage("puzzle-a-day-pieces.png"))
The puzzle-a-day board and pieces
In this puzzle, you are given the pieces below
and asked to place them on the board to cover everything except
today's date. For example, on 22 July, you could place the pieces like this:
![](javascript:showlimage("puzzle-a-day-22-july.png"))
A solution of puzzle-a-day for 22 July
DragonFjord make and sell wooden and plastic versions of puzzle-a-day, which
you can buy from Maths Gear—who
also provide the top prize for the 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.
![](javascript:showlimage("puzzle-a-day-31-september.png"))
A solution of puzzle-a-day for 31 September?!
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 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 this board and these pieces:
![](javascript:showlimage("new-puzzle-a-day-board.png"))
![](javascript:showlimage("new-puzzle-a-day-pieces.png"))
The board and pieces for the new puzzle
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...
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