I need help figuring out this math puzzle: I have a $11\times13\text{ cm}$ rectangle and I need help figuring out the least number of squares I need to cover the rectangle without overlap. I'm told the answer should be at most 5. If you can, provide a picture to help me understand.
[Math] the fewest number of squares required to cover a $11\times13\text{ cm}$ rectangle without overlap
recreational-mathematics
Related Solutions
I assume that (as usual in this type of problem) the operations are executed left toright (not by arithemtic priority). Here are the solutions:
- [9, 4, 8, 6, 7, 3, 1, 2, 5]: $9+13=22$, $22\times 4=88$, $88:8=11$, $11+6=17$, $17+12=29$, $29\times 7=203$, $203-3=200$, $200-11=189$, $189+1=190$, $190\times 2=380$, $380:5=76$, $76-10=66$.
- [9, 7, 2, 8, 4, 3, 6, 1, 5]
- [9, 6, 3, 8, 5, 7, 2, 1, 4]
- [9, 1, 4, 8, 2, 7, 5, 6, 3]
- [9, 5, 3, 8, 1, 2, 7, 6, 4]
- [9, 2, 4, 7, 8, 6, 5, 1, 3]
- [9, 3, 2, 4, 8, 7, 6, 1, 5]
- [9, 3, 2, 7, 6, 5, 8, 1, 4]
- [9, 5, 3, 7, 1, 2, 8, 6, 4]
- [9, 4, 3, 5, 2, 1, 8, 6, 7]
- [9, 3, 2, 6, 1, 7, 5, 8, 4]
- [9, 3, 2, 5, 1, 7, 6, 8, 4]
- [8, 9, 7, 1, 4, 2, 5, 3, 6]
- [7, 9, 8, 5, 6, 2, 4, 1, 3]
- [7, 9, 4, 8, 6, 2, 3, 1, 5]
- [7, 9, 5, 8, 3, 6, 1, 2, 4]
- [2, 9, 4, 3, 8, 6, 7, 1, 5]
- [7, 9, 5, 6, 1, 8, 3, 4, 2]
- [7, 9, 5, 3, 1, 8, 6, 4, 2]
- [1, 9, 7, 5, 2, 8, 6, 4, 3]
- [1, 9, 7, 3, 5, 8, 6, 2, 4]
- [3, 9, 6, 5, 4, 8, 7, 1, 2]
- [6, 9, 3, 1, 4, 8, 5, 2, 7]
- [1, 9, 3, 7, 2, 5, 8, 4, 6]
- [1, 9, 6, 2, 3, 7, 8, 4, 5]
- [4, 9, 3, 2, 6, 7, 8, 1, 5]
- [3, 9, 4, 6, 5, 1, 8, 2, 7]
- [2, 9, 5, 7, 1, 3, 6, 8, 4]
- [2, 9, 5, 6, 1, 3, 7, 8, 4]
- [7, 9, 5, 6, 3, 1, 2, 4, 8]
- [1, 9, 7, 2, 5, 3, 6, 4, 8]
- [8, 6, 9, 7, 5, 3, 1, 2, 4]
- [8, 3, 9, 5, 7, 6, 1, 2, 4]
- [8, 6, 9, 5, 2, 1, 7, 4, 3]
- [8, 5, 9, 3, 6, 4, 7, 1, 2]
- [8, 1, 9, 3, 4, 2, 7, 6, 5]
- [8, 6, 9, 3, 5, 2, 1, 4, 7]
- [8, 5, 9, 1, 4, 2, 3, 6, 7]
- [5, 4, 9, 1, 8, 7, 2, 3, 6]
- [5, 4, 9, 1, 6, 8, 7, 2, 3]
- [5, 4, 9, 3, 7, 6, 8, 1, 2]
- [5, 1, 9, 3, 7, 2, 8, 4, 6]
- [5, 6, 9, 7, 1, 4, 3, 8, 2]
- [5, 4, 9, 7, 1, 3, 6, 8, 2]
- [5, 6, 9, 3, 1, 4, 7, 8, 2]
- [5, 4, 9, 6, 1, 3, 7, 8, 2]
- [5, 1, 9, 2, 3, 6, 7, 8, 4]
- [5, 4, 9, 1, 3, 2, 7, 8, 6]
- [5, 1, 9, 2, 4, 3, 7, 8, 6]
- [7, 3, 9, 4, 5, 2, 1, 6, 8]
- [5, 7, 9, 1, 6, 2, 3, 4, 8]
- [2, 6, 9, 1, 7, 3, 5, 4, 8]
- [8, 1, 7, 9, 2, 4, 5, 6, 3]
- [8, 4, 3, 9, 5, 7, 1, 2, 6]
- [8, 1, 3, 9, 5, 2, 6, 4, 7]
- [2, 8, 4, 9, 1, 7, 5, 6, 3]
- [5, 8, 6, 9, 1, 3, 7, 4, 2]
- [7, 6, 8, 9, 2, 5, 1, 4, 3]
- [5, 2, 4, 9, 8, 7, 6, 1, 3]
- [3, 1, 4, 9, 8, 6, 7, 2, 5]
- [7, 1, 3, 9, 6, 8, 5, 2, 4]
- [7, 1, 4, 9, 2, 8, 5, 6, 3]
- [5, 7, 6, 9, 4, 8, 3, 1, 2]
- [1, 7, 6, 9, 3, 8, 2, 4, 5]
- [5, 1, 6, 9, 7, 8, 3, 2, 4]
- [5, 4, 1, 9, 3, 8, 6, 2, 7]
- [4, 6, 3, 9, 7, 2, 8, 1, 5]
- [5, 1, 3, 9, 6, 7, 8, 2, 4]
- [5, 4, 3, 9, 6, 1, 8, 2, 7]
- [7, 3, 6, 9, 1, 5, 4, 8, 2]
- [7, 3, 4, 9, 2, 5, 1, 8, 6]
- [2, 7, 6, 9, 1, 4, 5, 8, 3]
- [2, 6, 3, 9, 1, 7, 5, 8, 4]
- [5, 3, 2, 9, 1, 6, 7, 8, 4]
- [7, 1, 3, 9, 6, 5, 2, 4, 8]
- [7, 3, 1, 9, 2, 5, 6, 4, 8]
- [8, 1, 7, 3, 9, 5, 6, 2, 4]
- [8, 4, 7, 3, 9, 5, 1, 2, 6]
- [8, 4, 6, 5, 9, 3, 1, 2, 7]
- [7, 4, 8, 5, 9, 6, 2, 1, 3]
- [7, 2, 8, 1, 9, 4, 5, 3, 6]
- [3, 4, 8, 7, 9, 5, 1, 2, 6]
- [7, 2, 6, 8, 9, 5, 4, 1, 3]
- [7, 1, 3, 8, 9, 5, 4, 2, 6]
- [3, 6, 4, 8, 9, 7, 2, 1, 5]
- [3, 1, 6, 8, 9, 7, 4, 2, 5]
- [7, 6, 4, 2, 9, 8, 3, 1, 5]
- [3, 7, 6, 5, 9, 8, 2, 1, 4]
- [1, 5, 7, 4, 9, 3, 8, 2, 6]
- [5, 1, 6, 3, 9, 7, 8, 2, 4]
- [3, 2, 6, 1, 9, 7, 5, 4, 8]
- [2, 1, 5, 3, 9, 6, 7, 4, 8]
- [1, 8, 6, 7, 3, 9, 2, 4, 5]
- [6, 8, 4, 5, 3, 9, 7, 1, 2]
- [7, 5, 8, 4, 2, 9, 1, 6, 3]
- [7, 2, 5, 1, 8, 9, 4, 3, 6]
- [1, 2, 7, 5, 8, 9, 4, 3, 6]
- [3, 5, 2, 7, 8, 9, 4, 1, 6]
- [1, 4, 7, 5, 2, 9, 8, 6, 3]
- [1, 4, 3, 7, 2, 9, 8, 6, 5]
- [2, 1, 5, 3, 7, 9, 8, 4, 6]
- [2, 7, 3, 6, 1, 9, 5, 8, 4]
- [2, 7, 3, 5, 1, 9, 6, 8, 4]
- [3, 7, 2, 5, 1, 9, 4, 8, 6]
- [3, 7, 2, 4, 1, 9, 5, 8, 6]
- [1, 4, 7, 5, 3, 9, 2, 8, 6]
- [6, 3, 4, 7, 2, 9, 1, 8, 5]
- [4, 1, 6, 7, 3, 9, 2, 8, 5]
- [5, 1, 6, 2, 3, 9, 7, 8, 4]
- [2, 5, 6, 4, 3, 9, 1, 8, 7]
- [6, 2, 1, 5, 3, 9, 7, 4, 8]
- [8, 1, 7, 5, 6, 4, 9, 2, 3]
- [5, 8, 6, 7, 1, 3, 9, 4, 2]
- [2, 8, 4, 5, 1, 7, 9, 6, 3]
- [1, 7, 8, 6, 4, 5, 9, 2, 3]
- [3, 6, 8, 7, 5, 1, 9, 2, 4]
- [1, 6, 7, 8, 2, 5, 9, 4, 3]
- [2, 1, 5, 8, 6, 3, 9, 4, 7]
- [2, 1, 6, 3, 8, 5, 9, 4, 7]
- [2, 6, 5, 4, 7, 8, 9, 1, 3]
- [7, 3, 6, 4, 1, 5, 9, 8, 2]
- [7, 2, 5, 1, 3, 4, 9, 8, 6]
- [2, 7, 6, 5, 1, 4, 9, 8, 3]
- [1, 3, 7, 5, 2, 6, 9, 8, 4]
- [1, 2, 7, 5, 3, 4, 9, 8, 6]
- [5, 3, 2, 7, 1, 6, 9, 8, 4]
- [2, 6, 3, 5, 1, 7, 9, 8, 4]
- [5, 2, 4, 1, 3, 7, 9, 8, 6]
- [3, 5, 4, 1, 2, 7, 9, 8, 6]
- [4, 5, 1, 7, 3, 6, 9, 2, 8]
- [7, 4, 6, 8, 1, 2, 5, 9, 3]
- [7, 4, 6, 5, 1, 2, 8, 9, 3]
- [8, 7, 3, 1, 2, 4, 5, 6, 9]
- [8, 2, 1, 7, 3, 6, 5, 4, 9]
- [8, 2, 1, 6, 3, 5, 7, 4, 9]
- [1, 8, 7, 2, 6, 3, 5, 4, 9]
- [1, 8, 4, 7, 5, 2, 6, 3, 9]
- [5, 8, 3, 1, 2, 4, 7, 6, 9]
- [3, 7, 8, 5, 4, 1, 2, 6, 9]
- [7, 1, 2, 8, 6, 3, 5, 4, 9]
- [7, 2, 1, 8, 4, 6, 5, 3, 9]
- [1, 4, 2, 8, 5, 7, 6, 3, 9]
- [3, 1, 4, 8, 5, 2, 7, 6, 9]
- [5, 7, 4, 1, 8, 6, 3, 2, 9]
- [2, 1, 3, 6, 8, 7, 5, 4, 9]
- [7, 1, 2, 4, 5, 8, 3, 6, 9]
- [4, 6, 1, 7, 2, 8, 5, 3, 9]
- [1, 2, 4, 7, 5, 8, 3, 6, 9]
- [2, 3, 5, 6, 7, 8, 1, 4, 9]
- [4, 6, 1, 5, 2, 7, 8, 3, 9]
- [2, 7, 6, 4, 3, 5, 1, 8, 9]
- [2, 1, 6, 4, 5, 3, 7, 8, 9]
First of all, scale down the various ingredient requirements from a 48-cookie batch size to an individual cookie size. This is done by dividing your three amounts {1.5, 1, 2.75} by 48, giving {1/32, 1/48, 11/192}. Then all one has to do is multiply the required cookie batch size by each of these individual amounts. At least that is how I'd do it.
Best Answer
I can prove there is no 5-square solution. The partitions of $11\times 13 = 143$ into sums of five squares can be enumerated: $$ \matrix{1^2 &+ 1^2 &+ 2^2 &+ 4^2 &+ 11^2\cr 1^2 &+ 1^2 &+ 4^2 &+ 5^2 &+ 10^2\cr 1^2 &+ 2^2 &+ 5^2 &+ 7^2 &+ 8^2\cr 1^2 &+ 3^2 &+ 4^2 &+ 6^2 &+ 9^2\cr 2^2 &+ 4^2 &+ 5^2 &+ 7^2 &+ 7^2\cr 2^2 &+ 5^2 &+ 5^2 &+ 5^2 &+ 8^2\cr 3^2 &+ 3^2 &+ 3^2 &+ 4^2 &+ 10^2\cr 3^2 &+ 3^2 &+ 5^2 &+ 6^2 &+ 8^2\cr }$$ All but one of these can be eliminated out of hand, by looking at the two largest squares: $a \times a$ and $b \times b$ squares can't fit in a rectangle without overlapping unless the rectangle has one dimension at least $a+b$. The remaining possiblility is $2^2 + 5^2 + 5^2 + 5^2 + 8^2$, but it's easy to see that an $8 \times 8$ square and three $5 \times 5$ squares won't fit in the rectangle.
EDIT: There's no tiling of the $11 \times 13$ rectangle with $5$ squares even if you don't require integer sides. It's best to work up to $5$ tiles one at a time.
With one tile ($a \times a$) you can only tile an $a \times a$ rectangle.
With two tiles, both must be $a \times a$, and you get an $a \times 2a$ rectangle. Henceforth, I'll leave out the $a$, and assume the greatest common divisor of edge lengths is $1$, so call this $1 \times 2$.
With three tiles, at least one must be on an edge of your rectangle, and the rest of the rectangle is a two-tile rectangle. There are two cases, depending on how that two-tile rectangle is oriented:
With four tiles, you could put another square on one side of a three-tile rectangle, or you could have four equal squares, each taking one corner of a $2 \times 2$ square. There are five possibilities.
With five tiles, you could put one square on one side of a four-tile rectangle, obtaining a $4 \times 7$, $7 \times 3$, $5 \times 1$, $4 \times 5$, $4 \times 2$, $8 \times 5$, $3 \times 8$, $7 \times 2$ or $5 \times 7$ rectangle. Or if no square takes a whole side of the rectangle, you must have one square in each of the four corners of the rectangle and one square not on a corner. If so, it's not hard to see that this non-corner square must be on an edge, let's say the right edge. On the left edge, the two squares may be the same size (resulting in a $6 \times 5$ rectangle) or different sizes. If they are different sizes, the smaller one must be the same size as its neighbour to the right, resulting in a $7 \times 6$ rectangle.
(I hope) that's all the possibilities, not counting rotations and reflections. None of the possibilities has an $11$ to $13$ ratio.