Problem 1

In the plane, points with integer coordinates are vertices of unit squares, colored alternately black and white as on a chessboard. For positive integers

m,n

, consider a right triangle with integer-coordinate vertices and legs of lengths

m,n

along square edges. Let

S_1

be the black area and

S_2

the white area. Define

f(m,n)=|S_1-S_2|

. (a) Calculate

f(m,n)

when

m,n

are both even or both odd. (b) Prove

f(m,n)\le\max(m,n)/2

. (c) Show that no constant

C

satisfies

f(m,n)<C

for all

m,n

Step 3 of 3: Construct unbounded excess

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In plain words

Construct unbounded excess

f(n+1,n)=\dfrac{n-1}{6}

Detailed analysis

Take even $n$ and compare the triangles with legs $(n,n)$ and $(n+1,n)$ . The first has zero excess. Along the added strip, summing the similar white triangles gives white area $(2n+1)/12$ ; the black area is $n/2-(2n+1)/12$ , so the new excess is $(n-1)/6$ , which tends to infinity.