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Difference between revisions of "2009 USAMO Problems/Problem 3"

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== Problem ==
 
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We define a ''chessboard polygon'' to be a polygon whose sides are situated along lines of the form <math>x = a</math> or <math>y = b</math>, where <math>a</math> and <math>b</math> are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping <math>1 \times 2</math> rectangles. Finally, a ''tasteful tiling'' is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a <math>3 \times 4</math> rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.
 
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<center><asy>
 
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size(400); pathpen = linewidth(2.5);
 
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void chessboard(int a, int b, pair P){
 
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for(int i = 0; i < a; ++i) for(int j = 0; j < b; ++j)
 
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  if((i+j) % 2 == 1) fill(shift(P.x+i,P.y+j)*unitsquare,rgb(0.6,0.6,0.6));
 
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D(P--P+(a,0)--P+(a,b)--P+(0,b)--cycle);
 
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}
 
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chessboard(2,2,(2.5,0));fill(unitsquare,rgb(0.6,0.6,0.6));fill(shift(1,1)*unitsquare,rgb(0.6,0.6,0.6)); chessboard(4,3,(6,0)); chessboard(4,3,(11,0)); MP("\mathrm{Distasteful\ tilings}",(2.25,3),fontsize(12));
 
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/* draw lines */
 
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D((0,0)--(2,0)--(2,2)--(0,2)--cycle); D((1,0)--(1,2)); D((2.5,1)--(4.5,1)); D((7,0)--(7,2)--(6,2)--(10,2)--(9,2)--(9,0)--(9,1)--(7,1)); D((8,2)--(8,3)); D((12,0)--(12,2)--(11,2)--(13,2)); D((13,1)--(15,1)--(14,1)--(14,3)); D((13,0)--(13,3));
 
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</asy></center> a) Prove that if a chessboard polygon can be tiled by dominoes, then it can be done so tastefully.
 
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b) Prove that such a tasteful tiling is unique.
 
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== Problem ==
 
== Problem ==
 
We define a ''chessboard polygon'' to be a polygon whose sides are situated along lines of the form <math>x = a</math> or <math>y = b</math>, where <math>a</math> and <math>b</math> are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping <math>1 \times 2</math> rectangles. Finally, a ''tasteful tiling'' is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a <math>3 \times 4</math> rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.
 
We define a ''chessboard polygon'' to be a polygon whose sides are situated along lines of the form <math>x = a</math> or <math>y = b</math>, where <math>a</math> and <math>b</math> are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping <math>1 \times 2</math> rectangles. Finally, a ''tasteful tiling'' is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a <math>3 \times 4</math> rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.
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== Solution ==
 
== Solution ==
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'''NOTE: this only covers part (a).'''
 
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First, tile the board in any way. This tiling may be tasteful or distasteful. If it is tasteful, we are done. But if it is ''dis''tasteful, look for the distasteful 2x2 arrays in the shape. Then, if the dominoes are vertical, make them horizontal, and they will now be tasteful. If they are horizontal, make them vertical, and they too will now be tasteful. <math>\blacksquare</math>
 
  
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{{solution}}
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{{Solution}}
  
 
== See also ==
 
== See also ==

Latest revision as of 18:51, 6 May 2023

Problem

We define a chessboard polygon to be a polygon whose sides are situated along lines of the form $x = a$ or $y = b$, where $a$ and $b$ are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping $1 \times 2$ rectangles. Finally, a tasteful tiling is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a $3 \times 4$ rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.

[asy] size(400); pathpen = linewidth(2.5); void chessboard(int a, int b, pair P){ for(int i = 0; i < a; ++i) for(int j = 0; j < b; ++j) if((i+j) % 2 == 1) fill(shift(P.x+i,P.y+j)*unitsquare,rgb(0.6,0.6,0.6)); D(P--P+(a,0)--P+(a,b)--P+(0,b)--cycle); } chessboard(2,2,(2.5,0));fill(unitsquare,rgb(0.6,0.6,0.6));fill(shift(1,1)*unitsquare,rgb(0.6,0.6,0.6)); chessboard(4,3,(6,0)); chessboard(4,3,(11,0)); MP("\mathrm{Distasteful\ tilings}",(2.25,3),fontsize(12)); /* draw lines */ D((0,0)--(2,0)--(2,2)--(0,2)--cycle); D((1,0)--(1,2)); D((2.5,1)--(4.5,1)); D((7,0)--(7,2)--(6,2)--(10,2)--(9,2)--(9,0)--(9,1)--(7,1)); D((8,2)--(8,3)); D((12,0)--(12,2)--(11,2)--(13,2)); D((13,1)--(15,1)--(14,1)--(14,3)); D((13,0)--(13,3)); [/asy]

a) Prove that if a chessboard polygon can be tiled by dominoes, then it can be done so tastefully.

b) Prove that such a tasteful tiling is unique.

Solution

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See also

2009 USAMO (Problems • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

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