3
$\begingroup$

Find the solutions of $|x|+|y| \le 10$, where $x$ and $y$ are integers.

My solution:

$$|x|+|y|+z=10$$

Now the solutions if there were no absolute values is:

$$\binom{13-1}{10}=\frac{11*12}{2}=66$$

now subtract that once that have $0$ then multiply the others by $4$ and multiply that ones that only has $1$ zeroes by $2$.I get $221$ Am I right?

$\endgroup$
  • 1
    $\begingroup$ Do you want the solutions, or the number of solutions? $\endgroup$ – Arthur Aug 4 '16 at 11:09
  • $\begingroup$ @Arthur number of the solutions. $\endgroup$ – Taha Akbari Aug 4 '16 at 11:09
  • $\begingroup$ i would start with $$x=-10,y=0$$ or $$x=-9,y=-1$$ or $$x=-9,y=0$$ or $$x=-9,y=1$$ and so on $\endgroup$ – Dr. Sonnhard Graubner Aug 4 '16 at 11:10
  • $\begingroup$ are you interested on all solutions? $\endgroup$ – Dr. Sonnhard Graubner Aug 4 '16 at 11:14
  • $\begingroup$ There is a nice graphical approach. $|x| + |y| \le 10$ describes the interior of a square with vertices $(0,\pm10), (\pm10,0)$. Simply shade in the integer points $(x,y)$ lying within this square. $\endgroup$ – Mr. Chip Aug 4 '16 at 11:15
5
$\begingroup$

The number of integer solutions of $|x|+|y|\le n$ is $$1+4n+4\sum_{k=1}^{n-1} k=2n(n+1)+1.$$ $1$ is for the origin, $4n$ is for the points on the 4 semi-axis, and finally $4\sum_{k=1}^{n-1} k$ counts the points in the 4 right triangles inside the 4 quadrants.

See also the Sloane' sequence https://oeis.org/A001844

$\endgroup$
2
$\begingroup$

Pick's Theorem says that the area of a polygon whose vertices have integer coordinates is given by

$$A=I+{B\over2}-1$$

where $I$ is the number of Interior points with integer coordinates and $B$ is the number of Boundary points with integer coordinates. For the given problem the polygon is a square with diagonals of length $20$, so $A=200$. The bounday points satisfy $|x|+|y|=10$, so it's easy to see that $B=40$, hence $I=200-{40\over2}+1=181$. The number we want is

$$I+B=181+40=221$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.