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Difference between revisions of "2023 AMC 8 Problems/Problem 2"

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==Problem==
 
==Problem==
 
A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
 
A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
−
[[Image:2023 AMC 8-2.png|thumb|center|400px]]
+
<asy>
 +
//Restored original diagram. Alter it if you would like, but it was made by TheMathGuyd,
 +
// Diagram by TheMathGuyd. I even put the lined texture :)
 +
// Thank you Kante314 for inspiring thicker arrows. They do look much better
 +
size(0,3cm);
 +
path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
 +
path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle;
 +
path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle;
 +
path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle;
 +
path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle;
 +
path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle;
 +
path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle;
 +
filldraw(sq,mediumgrey,black);
 +
draw((0.75,0)--(1.25,0),currentpen+1,Arrow(size=6));
 +
//folding
 +
path sqside = (-0.5,-0.5)--(0.5,-0.5);
 +
path rhside = (-0.125,-0.125)--(0.5,-0.5);
 +
transform fld = shift((1.75,0))*scale(0.5);
 +
draw(fld*sq,black);
 +
int i;
 +
for(i=0; i<10; i=i+1)
 +
{
 +
  draw(shift(0,0.05*i)*fld*sqside,deepblue);
 +
}
 +
path rhedge = (-0.125,-0.125)--(-0.125,0.8)--(-0.2,0.85)--cycle;
 +
filldraw(fld*rhedge,grey);
 +
path sqedge = (-0.5,-0.5)--(-0.5,0.4475)--(-0.575,0.45)--cycle;
 +
filldraw(fld*sqedge,grey);
 +
filldraw(fld*rh,white,black);
 +
int i;
 +
for(i=0; i<10; i=i+1)
 +
{
 +
  draw(shift(0,0.05*i)*fld*rhside,deepblue);
 +
}
 +
draw((2.25,0)--(2.75,0),currentpen+1,Arrow(size=6));
 +
//cutting
 +
transform cut = shift((3.25,0))*scale(0.5);
 +
draw(shift((-0.01,+0.01))*cut*sq);
 +
draw(cut*sq);
 +
filldraw(shift((0.01,-0.01))*cut*sq,white,black);
 +
int j;
 +
for(j=0; j<10; j=j+1)
 +
{
 +
draw(shift(0,0.05*j)*cut*sqside,deepblue);
 +
}
 +
draw(shift((0.01,-0.01))*cut*(0,-0.5)--shift((0.01,-0.01))*cut*(0.5,0),dashed);
 +
//Answers Below, but already Separated
 +
//filldraw(sqA,grey,black);
 +
//filldraw(sqB,grey,black);
 +
//filldraw(sq,grey,black);
 +
//filldraw(sqC,white,black);
 +
//filldraw(sq,grey,black);
 +
//filldraw(trD,white,black);
 +
//filldraw(sq,grey,black);
 +
//filldraw(sqE,white,black);
 +
</asy>
  
−
==Solution 1==
+
<asy>
−
Take a paper and fold it using the given conditions to see the resulting answer
+
// Diagram by TheMathGuyd.
−
-apex304
+
size(0,7.5cm);
 +
path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
 +
path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle;
 +
path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle;
 +
path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle;
 +
path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle;
 +
path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle;
 +
path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle;
  
−
==Solution 2==
+
//ANSWERS
−
Notice how the paper is folded. The bottom right corner of the twice-folded paper has to be the middle of the unfolded paper. So if you cut it in the way that it is shown in the problem, you find (it has to be symmetrical) that the cuts make a square centered in the middle of the paper.  
+
real sh = 1.5;
 +
label("$\textbf{(A)}$",(-0.5,0.5),SW);
 +
label("$\textbf{(B)}$",shift((sh,0))*(-0.5,0.5),SW);
 +
label("$\textbf{(C)}$",shift((2sh,0))*(-0.5,0.5),SW);
 +
label("$\textbf{(D)}$",shift((0,-sh))*(-0.5,0.5),SW);
 +
label("$\textbf{(E)}$",shift((sh,-sh))*(-0.5,0.5),SW);
 +
filldraw(sqA,mediumgrey,black);
 +
filldraw(shift((sh,0))*sqB,mediumgrey,black);
 +
filldraw(shift((2*sh,0))*sq,mediumgrey,black);
 +
filldraw(shift((2*sh,0))*sqC,white,black);
 +
filldraw(shift((0,-sh))*sq,mediumgrey,black);
 +
filldraw(shift((0,-sh))*trD,white,black);
 +
filldraw(shift((sh,-sh))*sq,mediumgrey,black);
 +
filldraw(shift((sh,-sh))*sqE,white,black);
 +
</asy>
  
−
-claregu
+
==Solution==
  
−
==Solution 3 (Thorough)==
+
Notice that when we unfold the paper along the vertical fold line, we get the following shape:
  
−
Notice that when we unfold the paper from the vertical fold line, we get
 
  
−
[[Image:Screenshot_2023-01-25_8.11.20_AM.png|thumb|center|200px]]
+
[working version of the diagram (harshu13)]:
 +
<asy>
  
−
Then, if we unfold the paper from the horizontal fold line, we result in
+
size(90);
 +
path sq = (-0.5,0)--(0.5,0)--(0.5,0.5)--(-0.5,0.5)--cycle;
 +
path trE = (-0.25,0)--(0.25,0)--(0,0.25)--cycle;
  
−
[[Image:Screenshot_2023-01-25_8.14.41_AM.png|thumb|center|200px]]
+
real sh = 1.5;
 +
filldraw(shift((sh,-sh))*sq,mediumgrey,black);
 +
filldraw(shift((sh,-sh))*trE,white,black);
  
−
It is clear that the answer is <math>\boxed{\textbf{(E)}}</math>
+
size(90);
 +
path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
 +
path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle;
  
 +
real sh = 1.5;
 +
filldraw(shift((sh,-sh))*sq,mediumgrey,black);
 +
filldraw(shift((sh,-sh))*sqE,white,black);
 +
</asy>
 +
 +
It is clear that the answer is <math>\boxed{\textbf{(E)}}</math>.
 
~MrThinker
 
~MrThinker
 +
 +
==*Easy Video Explanation by MathTalks_Now*==
 +
https://studio.youtube.com/video/PMOeiGLkDH0/edit
 +
 +
 +
 +
 +
 +
 +
==Video Solution by Math-X (Smart and Simple)==
 +
https://youtu.be/Ku_c1YHnLt0?si=ZucTBcN42MKGX2Ty&t=115 ~Math-X
 +
 +
==Video Solution (How to Creatively THINK!!!)==
 +
https://youtu.be/suFxwnH-ak8
 +
~Education the Study of everything
 +
 +
==Video Solution by Magic Square==
 +
https://youtu.be/-N46BeEKaCQ?t=5658
 +
 +
==Video Solution by SpreadTheMathLove==
 +
https://www.youtube.com/watch?v=EcrktBc8zrM
 +
==Video Solution by Interstigation==
 +
https://youtu.be/DBqko2xATxs&t=67
 +
 +
==Video Solution by WhyMath==
 +
 +
~savannahsolver
 +
 +
==Video Solution by harungurcan==
 +
https://www.youtube.com/watch?v=35BW7bsm_Cg&t=97s
 +
 +
~harungurcan
  
 
==See Also==
 
==See Also==
−
{{AMC8 box|year=2023|before=1|num-a=3}}
+
{{AMC8 box|year=2023|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 20:39, 15 January 2024

Problem

A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures? [asy] //Restored original diagram. Alter it if you would like, but it was made by TheMathGuyd, // Diagram by TheMathGuyd. I even put the lined texture :) // Thank you Kante314 for inspiring thicker arrows. They do look much better size(0,3cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; filldraw(sq,mediumgrey,black); draw((0.75,0)--(1.25,0),currentpen+1,Arrow(size=6)); //folding path sqside = (-0.5,-0.5)--(0.5,-0.5); path rhside = (-0.125,-0.125)--(0.5,-0.5); transform fld = shift((1.75,0))*scale(0.5); draw(fld*sq,black); int i; for(i=0; i<10; i=i+1) { draw(shift(0,0.05*i)*fld*sqside,deepblue); } path rhedge = (-0.125,-0.125)--(-0.125,0.8)--(-0.2,0.85)--cycle; filldraw(fld*rhedge,grey); path sqedge = (-0.5,-0.5)--(-0.5,0.4475)--(-0.575,0.45)--cycle; filldraw(fld*sqedge,grey); filldraw(fld*rh,white,black); int i; for(i=0; i<10; i=i+1) { draw(shift(0,0.05*i)*fld*rhside,deepblue); } draw((2.25,0)--(2.75,0),currentpen+1,Arrow(size=6)); //cutting transform cut = shift((3.25,0))*scale(0.5); draw(shift((-0.01,+0.01))*cut*sq); draw(cut*sq); filldraw(shift((0.01,-0.01))*cut*sq,white,black); int j; for(j=0; j<10; j=j+1) { draw(shift(0,0.05*j)*cut*sqside,deepblue); } draw(shift((0.01,-0.01))*cut*(0,-0.5)--shift((0.01,-0.01))*cut*(0.5,0),dashed); //Answers Below, but already Separated //filldraw(sqA,grey,black); //filldraw(sqB,grey,black); //filldraw(sq,grey,black); //filldraw(sqC,white,black); //filldraw(sq,grey,black); //filldraw(trD,white,black); //filldraw(sq,grey,black); //filldraw(sqE,white,black); [/asy]

[asy] // Diagram by TheMathGuyd. size(0,7.5cm); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path rh = (-0.125,-0.125)--(0.5,-0.5)--(0.5,0.5)--(-0.125,0.875)--cycle; path sqA = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,-0.25)--(0.25,0)--(0.5,0.25)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--(-0.5,0.25)--(-0.25,0)--(-0.5,-0.25)--cycle; path sqB = (-0.5,-0.5)--(-0.25,-0.5)--(0,-0.25)--(0.25,-0.5)--(0.5,-0.5)--(0.5,0.5)--(0.25,0.5)--(0,0.25)--(-0.25,0.5)--(-0.5,0.5)--cycle; path sqC = (-0.25,-0.25)--(0.25,-0.25)--(0.25,0.25)--(-0.25,0.25)--cycle; path trD = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; //ANSWERS real sh = 1.5; label("$\textbf{(A)}$",(-0.5,0.5),SW); label("$\textbf{(B)}$",shift((sh,0))*(-0.5,0.5),SW); label("$\textbf{(C)}$",shift((2sh,0))*(-0.5,0.5),SW); label("$\textbf{(D)}$",shift((0,-sh))*(-0.5,0.5),SW); label("$\textbf{(E)}$",shift((sh,-sh))*(-0.5,0.5),SW); filldraw(sqA,mediumgrey,black); filldraw(shift((sh,0))*sqB,mediumgrey,black); filldraw(shift((2*sh,0))*sq,mediumgrey,black); filldraw(shift((2*sh,0))*sqC,white,black); filldraw(shift((0,-sh))*sq,mediumgrey,black); filldraw(shift((0,-sh))*trD,white,black); filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*sqE,white,black); [/asy]

Solution

Notice that when we unfold the paper along the vertical fold line, we get the following shape:


[working version of the diagram (harshu13)]: [asy] size(90); path sq = (-0.5,0)--(0.5,0)--(0.5,0.5)--(-0.5,0.5)--cycle; path trE = (-0.25,0)--(0.25,0)--(0,0.25)--cycle; real sh = 1.5; filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*trE,white,black); size(90); path sq = (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle; path sqE = (-0.25,0)--(0,-0.25)--(0.25,0)--(0,0.25)--cycle; real sh = 1.5; filldraw(shift((sh,-sh))*sq,mediumgrey,black); filldraw(shift((sh,-sh))*sqE,white,black); [/asy]

It is clear that the answer is $\boxed{\textbf{(E)}}$. ~MrThinker

*Easy Video Explanation by MathTalks_Now*

https://studio.youtube.com/video/PMOeiGLkDH0/edit




Video Solution by Math-X (Smart and Simple)

https://youtu.be/Ku_c1YHnLt0?si=ZucTBcN42MKGX2Ty&t=115 ~Math-X

Video Solution (How to Creatively THINK!!!)

https://youtu.be/suFxwnH-ak8 ~Education the Study of everything

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=5658

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=EcrktBc8zrM

Video Solution by Interstigation

https://youtu.be/DBqko2xATxs&t=67

Video Solution by WhyMath

~savannahsolver

Video Solution by harungurcan

https://www.youtube.com/watch?v=35BW7bsm_Cg&t=97s

~harungurcan

See Also

2023 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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