MathDB

Bob proposes the following game to Johanna. The board in the figure is an equilateral triangle subdivided in turn into

256

small equilateral triangles, one of which is painted in black. Bob chooses any point inside the board and places a small token. Johanna can make three types of plays. Each of them consists of choosing any of the

3

vertices of the board and move the token to the midpoint between the current position of the tile and the chosen vertex. In the second figure we see an example of a move in which Johana chose vertex

A

. Johanna wins if she manages to place her piece inside the triangle black. Prove that Johanna can always win in at most

4

moves.
[asy] unitsize(8 cm);
pair A, B, C; int i;
A = dir(60); C = (0,0); B = (1,0);
fill((6/16*(1,0) + 1/16*dir(60))--(7/16*(1,0) + 1/16*dir(60))--(6/16*(1,0) + 2/16*dir(60))--cycle, gray(0.7)); draw(A--B--C--cycle);
for (i = 1; i <= 15; ++i) { draw(interp(A,B,i/16)--interp(A,C,i/16)); draw(interp(B,C,i/16)--interp(B,A,i/16)); draw(interp(C,A,i/16)--interp(C,B,i/16)); }
label("

A

", A, N); label("

B

", B, SE); label("

C

", C, SW); [/asy]
[asy] unitsize(8 cm);
pair A, B, C, X, Y, Z; int i;
A = dir(60); C = (0,0); B = (1,0); X = 9.2/16*(1,0) + 3.3/16*dir(60); Y = (A + X)/2; Z = rotate(60,X)*(Y);
fill((6/16*(1,0) + 1/16*dir(60))--(7/16*(1,0) + 1/16*dir(60))--(6/16*(1,0) + 2/16*dir(60))--cycle, gray(0.7)); draw(A--B--C--cycle);
for (i = 1; i <= 15; ++i) { draw(interp(A,B,i/16)--interp(A,C,i/16)); draw(interp(B,C,i/16)--interp(B,A,i/16)); draw(interp(C,A,i/16)--interp(C,B,i/16)); }
draw(A--X, dotted); draw(arc(Z,abs(X - Y),-12,40), Arrow(6));
label("

A

", A, N); label("

B

", B, SE); label("

C

", C, SW); dot(A); dot(X); dot(Y); [/asy]

Problem Statement