Perimeter-like Grids

Page 9


Perimeter and Semiperimeter

In general, the perimeter of a a×ba\times b grid is 2a+2b2a+2b, the semiperimeter of a a×ba\times b grid is a+ba+b.

For example, the perimeter of a 3×53\times 5 grid is 16, its semiperimeter is 8.

Perimeter-like Grids

If two grids have the same perimeter, then these 2 grids are "perimeter-like grids".

For example, 5×25\times 2 grid and 3×43\times 4 grid are perimeter-like grids.


Maximum number of unit squares

Suppose there's a grid, a+b=10a+b=10, if the number of unit squares in the grid reaches its maximum, then what is the value of aa and bb?

Let's write out the possibilities: 1×9,    2×8,    3×7,    4×6,    5×5,    6×4,    7×3,    8×2,    9×11\times9,~~~~2\times8,~~~~3\times7,~~~~4\times6,~~~~5\times5,~~~~6\times4,~~~~7\times3,~~~~8\times2,~~~~9\times1 The corresponding numbers of unit squares are: 9,      16,      21,      24,      25,      24,      21,      16,      99,~~~~~~16,~~~~~~21,~~~~~~24,~~~~~~25,~~~~~~24,~~~~~~21,~~~~~~16,~~~~~~9 We see that when a=b=5a=b=5, then the number of unit squares reaches its maximum.

We say that the maximum number of unit squares it can form when a+b=10a+b=10 is 25.

What if a+ba+b is an odd number? Let's try this out...

Suppose a+b=9a+b=9, the possibilities are: 1×8,    2×7,    3×6,    4×5,    5×4,    6×3,    7×2,    8×11\times8,~~~~2\times7,~~~~3\times6,~~~~4\times5,~~~~5\times4,~~~~6\times3,~~~~7\times2,~~~~8\times1 The corresponding numbers of unit squares are: 8,      14,      18,      20,      20,      18,      14,      88,~~~~~~14,~~~~~~18,~~~~~~20,~~~~~~20,~~~~~~18,~~~~~~14,~~~~~~8 We see that when a=b±1a=b\pm1, then the number of unit squares reaches its maximum.

We say that the maximum number of unit squares it can form when a+b=9a+b=9 is 20.


The Greatest Rule

If a+ba+b is a fixed even number, then when a=ba=b, the number of unit squares it can form in the grid will reach its maximum.

Another way of thinking about this is among all the perimeter-like grids which a+ba+b is even, then the grids which has the property of a=ba=b will have the most number of unit squares.

If a+ba+b is a fixed odd number, then when a=b±1a=b\pm1, the number of unit squares it can form in the grid will reach its maximum.

Another way of thinking about this is among all the perimeter-like grids which a+ba+b is odd, then the grids which has the property of a=b±1a=b\pm1 will have the most number of unit squares.

Proof:
Consider a+ba+b is even. Suppose a+b=2ka+b=2k, a=k+xa=k+x, b=kxb=k-x (kk is a positive integer while xx is an integer). The number of unit squares in the grid is a×b=(k+x)(kx)=k2x2\begin{aligned} a\times b&=(k+x)(k-x) \\&=k^2-x^2 \end{aligned} Since x20x^2\geqslant 0, to maximize the value of k2x2k^2-x^2, xx must be equal to 0, thus a=b=ka=b=k.
Consider a+ba+b is odd. Suppose a+b=2k+1a+b=2k+1, a=k+xa=k+x, b=k+1xb=k+1-x. The number of unit squares in the grid is a×b=(k+x+1)(kx)=k2+kx2x=k2+kx(x1)\begin{aligned} a\times b&=(k+x+1)(k-x) \\&=k^2+k-x^2-x \\&=k^2+k-x(x-1) \end{aligned} Since x(x1)0x(x-1)\geqslant 0, to maximize the value of k2+kx(x1)k^2+k-x(x-1), xx must be equal to 0 or 1, thus a=ka=k, b=k+1b=k+1 or a=k+1a=k+1, b=kb=k, a=b±1a=b\pm1.

Generalization:

If a+ba+b is a fixed even number, then when a=b=ka=b=k, the number of unit squares in the grid will reach its maximum, which is ab=k2=(a+b2)2ab=k^2=\left (\frac{a+b}{2} \right )^2.

\bullet Hence if a+ba+b is even, the maximum number of unit squares it can form is (a+b2)2\left (\frac{a+b}{2} \right )^2.

If a+ba+b is a fixed odd number, then when a=k+1a=k+1, b=kb=k or when a=ka=k, b=k+1b=k+1, the number of unit squares in the grid will reach its maximum, which is ab=k(k+1)=(2k2)(2k+22)=(a+b1)(a+b+1)4ab=k(k+1)=\left (\frac{2k}{2} \right )\left (\frac{2k+2}{2} \right )=\frac{(a+b-1)(a+b+1)}{4}.

\bullet Hence if a+ba+b is odd, the maximum number of unit squares it can form is (a+b1)(a+b+1)4\frac{(a+b-1)(a+b+1)}{4}.


Worked examples

If a+b=14a+b=14, what is the maximum number of unit squares it can form?

Solution:

Since 14 is even, according to the greatest rule, the maximum number of unit squares it can form is (a+b2)2=72=49\left( \frac{a+b}{2} \right)^2=7^2=49.


If a+b=19a+b=19, what is the maximum number of unit squares it can form?

Solution:

Since 19 is odd, according to the greatest rule, the maximum number of unit squares it can form is (a+b1)(a+b+1)4=18×204=90\frac{(a+b-1)(a+b+1)}{4}=\frac{18\times 20}{4}=90.


This is one part of Grids and Quadrilaterals.

Note by Kenneth Tan
5 years, 1 month ago

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