# Square Crayon Sticks

Ten sets of crayons forming all the digits from 0 to 9 can be moved, flipped, rotate and intersect without changing their digital forms. What is the maximum number of 1x1 stick squares that can be formed by combining all the crayon digits?

• can we move the individual crayons, or do we have to preserve the digit shapes?
– Bass
Commented Aug 21, 2021 at 6:37
• Their digital forms must be preserved . can move individually to reposition but all digits 0-9 shall be visible according to colors
– TSLF
Commented Aug 21, 2021 at 6:55

Here is one way of making the rectangular grid arrangement that Bass suggested:

Only the digit 2 in the middle is flipped over.

By the way, this kind of packing puzzle shape is called a Polystick.

• The other way making non-mirrored and non-rotated solution of the above is the one with diagonal symmetry of 7,0,2,9 only.
– TSLF
Commented Aug 28, 2021 at 20:43

If we can rearrange all the 49 crayons independently, then we can make this shape:

Since we know that any square shaped bunch of unit squares is automatically the optimal way to maximally overlap the unit square sides, and continuing from a square to a rectangle by adding a single row also gives maximal crayon overlap, we know that this pattern is an optimal solution, and the maximum number of unit squares it's possible to make by rearranging individual crayons is

twenty.

If we have to preserve the digit shapes, the answer will be something a bit different, and I'll have to get back to you later :-)

(EDIT: I'm afraid that this is probably the case, since everything seems to make a bit more sense if the "their" in "without changing their forms" refers to the sets instead of the crayons. Oh well.)