A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
in the array can be changed into either
![\lceil x\rceil](/media/m/e/5/5/e55d3cf23126e3a9283aa4ce0b473552.png)
or
![\lfloor x\rfloor](/media/m/f/7/3/f731119c18efc9ba5bad155f27328395.png)
so that the row-sums and column-sums remain unchanged. (Note that
![\lceil x\rceil](/media/m/e/5/5/e55d3cf23126e3a9283aa4ce0b473552.png)
is the least integer greater than or equal to
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
, while
![\lfloor x\rfloor](/media/m/f/7/3/f731119c18efc9ba5bad155f27328395.png)
is the greatest integer less than or equal to
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
.)
%V0
A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil$ or $\lfloor x\rfloor$ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil$ is the least integer greater than or equal to $x$, while $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$.)