perplexus.info :: Just Math : Square Plane Function

Square Plane Function (Posted on 2010-11-16)

Let f be a real-valued function on the plane such that for every square ABCD in the plane, f(A) + f(B) + f(C) + f(D) = 0.

Prove or disprove that f(P) = 0 for every point P in the plane?

Submitted by Bractals

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Solution: (Hide)

Let P be an arbitrary point in the plane.

Let KLMN be a square in the plane such that
P is the intersection of line segments KM
and LN.

Let Q, R, S, and T be the midpoints of sides
KL, LM, MN, and NK respectively.

Consider the six squares: KQPT, LRPQ, MSPR,
NTPS, KLMN, and QRST -
0 = [f(K) + f(Q) + f(P) + f(T)] + [f(L) + f(R) + f(P) + f(Q)] + [f(M) + f(S) + f(P) + f(R)] + [f(N) + f(T) + f(P) + f(S)] - [f(K) + f(L) + f(M) + f(N)] - 2 [f(Q) + f(R) + f(S) + f(T)] = 4 f(P)
Thus, f(P) = 0 for all points P in the plane.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Another way	Larry	2010-11-21 19:15:29
Cheating on the square	Steve Herman	2010-11-16 15:11:03
Squaring the square (spoiler)	Steve Herman	2010-11-16 15:06:44

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