$\text{ }$ [asy][asy] unitsize(11); for(int i=0; i<6; ++i) { if(i<5) draw( (i, 0)--(i,5) ); else draw( (i, 0)--(i,2) ); if(i < 3) draw((0,i)--(5,i)); else draw((0,i)--(4,i)); } [/asy][/asy] We are dividing the above figure into parts with shapes: [asy][asy] unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((0,0)--(1,0)); draw((0,1)--(2,1)); draw((0,2)--(2,2)); [/asy][/asy][asy][asy] unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((3,1)--(3,2)); draw((0,0)--(1,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); [/asy][/asy] After that division, find the number of [asy][asy] unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((0,0)--(1,0)); draw((0,1)--(2,1)); draw((0,2)--(2,2)); [/asy][/asy] shaped parts.
Problem
Source:
Tags: induction
19.01.2013 01:18
First, let me give you a nice problem:
Now, come back to our problem. Let $x$ be the number of shapes with three cells and let $y$ be the number of shapes with $4$ cells. So we have $3x + 4y = 22$. Solving this, you get $(x,y)=(6,1)$, or $(2,4)$. Checking the answer $(6, 1)$ you can easily draw this: [asy][asy] unitsize(11); for(int i=0; i<6; ++i) { if(i<5) draw( (i, 0)--(i,5) ); else draw( (i, 0)--(i,2)); if(i < 3) draw((0,i)--(5,i)); else draw((0,i)--(4,i)); } draw((2,0)--(5,0),red+ linewidth(2)); draw((5,0)--(5,2),red+ linewidth(2)); draw((4,2)--(5,2), red+linewidth(2)); draw((4,1)--(4,2),red+ linewidth(2)); draw((2,1)--(4,1), red+linewidth(2)); draw((2,0)--(2,1), red+linewidth(2)); fill(shift(0,0 )*unitsquare); fill(shift(1,0 )*unitsquare); fill(shift(0,1 )*unitsquare); [/asy][/asy] Now, you have a $4\times 4$ square with one cell removed and according to the above problem, you can cover it with those three cells shapes. So the answer is $6$. Here's a covering: [asy][asy] unitsize(11); for(int i=0; i<6; ++i) { if(i<5) draw( (i, 0)--(i,5) ); else draw( (i, 0)--(i,2)); if(i < 3) draw((0,i)--(5,i)); else draw((0,i)--(4,i)); } draw((2,0)--(5,0),red+ linewidth(2)); draw((5,0)--(5,2),red+ linewidth(2)); draw((4,2)--(5,2), red+linewidth(2)); draw((4,1)--(4,2),red+ linewidth(2)); draw((2,1)--(4,1), red+linewidth(2)); draw((2,0)--(2,1), red+linewidth(2)); draw((0,2)--(1,2),blue+ linewidth(2)); draw((1,2)--(1,1),blue+ linewidth(2)); draw((1,1)--(2,1),blue+linewidth(2)); draw((2,1)--(2,3),blue+ linewidth(2)); draw((2,3)--(0,3),blue+linewidth(2)); draw((0,3)--(0,2),blue+linewidth(2)); draw((2,1)--(4,1),green+ linewidth(2)); draw((4,1)--(4,3),green+ linewidth(2)); draw((4,3)--(3,3),green+linewidth(2)); draw((3,3)--(3,2),green+ linewidth(2)); draw((3,2)--(2,2),green+linewidth(2)); draw((2,2)--(2,1),green+linewidth(2)); draw((4,3)--(4,5),yellow+ linewidth(2)); draw((4,5)--(2,5),yellow+ linewidth(2)); draw((2,5)--(2,4),yellow+linewidth(2)); draw((2,4)--(3,4),yellow+ linewidth(2)); draw((3,4)--(3,3),yellow+linewidth(2)); draw((3,3)--(4,3),yellow+linewidth(2)); draw((0,3)--(0,5),purple+ linewidth(2)); draw((0,5)--(2,5),purple+ linewidth(2)); draw((2,5)--(2,4),purple+linewidth(2)); draw((2,4)--(1,4),purple+ linewidth(2)); draw((1,4)--(1,3),purple+linewidth(2)); draw((1,3)--(0,3),purple+linewidth(2)); draw((1,3)--(2,3),black+ linewidth(2)); draw((2,3)--(2,2),black+ linewidth(2)); draw((2,2)--(3,2),black+linewidth(2)); draw((3,2)--(3,4),black+ linewidth(2)); draw((3,4)--(1,4),black+linewidth(2)); draw((1,4)--(1,3),black+linewidth(2)); fill(shift(0,0 )*unitsquare); fill(shift(1,0 )*unitsquare); fill(shift(0,1 )*unitsquare); [/asy][/asy]
21.02.2016 22:37
(Sorry for not using LaTeX)Here's an example for x=2 and y=4:
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