Difference between revisions of "2023 AMC 8 Problems/Problem 4"
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==Problem== | ==Problem== | ||
− | The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7. How many of these four numbers are prime? | + | The numbers from <math>1</math> to <math>49</math> are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number <math>7.</math> How many of these four numbers are prime? |
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | size(175); | ||
+ | void ds(pair p) { | ||
+ | filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,mediumgrey); | ||
+ | } | ||
+ | ds((0.5,4.5)); | ||
+ | ds((1.5,3.5)); | ||
+ | ds((3.5,1.5)); | ||
+ | ds((4.5,0.5)); | ||
+ | |||
+ | add(grid(7,7,grey+linewidth(1.25))); | ||
+ | |||
+ | int adj = 1; | ||
+ | int curUp = 2; | ||
+ | int curLeft = 4; | ||
+ | int curDown = 6; | ||
+ | |||
+ | label("$1$",(3.5,3.5)); | ||
+ | |||
+ | for (int len = 3; len<=3; len+=2) | ||
+ | { | ||
+ | for (int i=1; i<=len-1; ++i) | ||
+ | { | ||
+ | label("$"+string(curUp)+"$",(3.5+adj,3.5-adj+i)); | ||
+ | label("$"+string(curLeft)+"$",(3.5+adj-i,3.5+adj)); | ||
+ | label("$"+string(curDown)+"$",(3.5-adj,3.5+adj-i)); | ||
+ | ++curDown; | ||
+ | ++curLeft; | ||
+ | ++curUp; | ||
+ | } | ||
+ | ++adj; | ||
+ | curUp = len^2 + 1; | ||
+ | curLeft = len^2 + len + 2; | ||
+ | curDown = len^2 + 2*len + 3; | ||
+ | } | ||
+ | |||
+ | draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2)); | ||
+ | </asy> | ||
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math> | <math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math> | ||
− | ==Solution== | + | ==Solution 1== |
+ | |||
+ | We fill out the grid, as shown below: | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | size(175); | ||
+ | |||
+ | void ds(pair p) { | ||
+ | filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,mediumgrey); | ||
+ | } | ||
+ | |||
+ | ds((0.5,4.5)); | ||
+ | ds((1.5,3.5)); | ||
+ | ds((3.5,1.5)); | ||
+ | ds((4.5,0.5)); | ||
+ | |||
+ | add(grid(7,7,grey+linewidth(1.25))); | ||
+ | |||
+ | int adj = 1; | ||
+ | int curUp = 2; | ||
+ | int curLeft = 4; | ||
+ | int curDown = 6; | ||
+ | int curRight = 8; | ||
+ | |||
+ | label("$1$",(3.5,3.5)); | ||
+ | |||
+ | for (int len = 3; len<=7; len+=2) | ||
+ | { | ||
+ | for (int i=1; i<=len-1; ++i) | ||
+ | { | ||
+ | label("$"+string(curUp)+"$",(3.5+adj,3.5-adj+i)); | ||
+ | label("$"+string(curLeft)+"$",(3.5+adj-i,3.5+adj)); | ||
+ | label("$"+string(curDown)+"$",(3.5-adj,3.5+adj-i)); | ||
+ | label("$"+string(curRight)+"$",(3.5-adj+i,3.5-adj)); | ||
+ | ++curDown; | ||
+ | ++curLeft; | ||
+ | ++curUp; | ||
+ | ++curRight; | ||
+ | } | ||
+ | ++adj; | ||
+ | curUp = len^2 + 1; | ||
+ | curLeft = len^2 + len + 2; | ||
+ | curDown = len^2 + 2*len + 3; | ||
+ | curRight = len^2 + 3*len + 4; | ||
+ | } | ||
+ | |||
+ | draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2)); | ||
+ | </asy> | ||
+ | From the four numbers that appear in the shaded squares, <math>\boxed{\textbf{(D)}\ 3}</math> of them are prime: <math>19,23,</math> and <math>47.</math> | ||
+ | |||
+ | ~MathFun1000, MRENTHUSIASM | ||
+ | |||
+ | ==Solution 2== | ||
+ | Note that given time constraint, it's better to only count from perfect squares (in pink), as shown below: | ||
+ | <asy> | ||
+ | /* Grid Made by MRENTHUSIASM */ | ||
+ | /* Squares pattern solution input by TheMathGuyd */ | ||
+ | size(175); | ||
+ | |||
+ | void ds(pair p) { | ||
+ | filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,mediumgrey); | ||
+ | } | ||
+ | |||
+ | void ps(pair p) { | ||
+ | filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,pink+opacity(0.3)); | ||
+ | } | ||
+ | real ts = 0.5; | ||
+ | |||
+ | ds((0.5,4.5));label("$39$",(0.5,4.5)); | ||
+ | ds((1.5,3.5));label("$19$",(1.5,3.5)); | ||
+ | ds((3.5,1.5));label("$23$",(3.5,1.5)); | ||
+ | ds((4.5,0.5));label("$47$",(4.5,0.5)); | ||
+ | |||
+ | ps((3.5,3.5));label("$1$",(3.5,3.5)); | ||
+ | ps((4.5,2.5));label("$9$",(4.5,2.5)); | ||
+ | ps((5.5,1.5));label("$25$",(5.5,1.5)); | ||
+ | ps((6.5,0.5));label("$49$",(6.5,0.5)); | ||
+ | ps((3.5,4.5));label("$4$",(3.5,4.5)); | ||
+ | ps((2.5,5.5));label("$16$",(2.5,5.5)); | ||
+ | ps((1.5,6.5));label("$36$",(1.5,6.5)); | ||
+ | label(scale(ts)*"$\leftarrow$",(1,6),NE); | ||
+ | label(scale(ts)*"$+1$",(1,6),NW); | ||
+ | label(scale(ts)*"$\downarrow$",(1,6),SW); | ||
+ | label(scale(ts)*"$+2$",(1,5),NW); | ||
+ | label(scale(ts)*"$\downarrow$",(1,5),SW); | ||
+ | label(scale(ts)*"$+3$",(1,4),NW); | ||
+ | label(scale(ts)*"$+1$",(2,5),NW); | ||
+ | label(scale(ts)*"$\downarrow$",(2,5),SW); | ||
+ | label(scale(ts)*"$+2$",(2,4),NW); | ||
+ | label(scale(ts)*"$\downarrow$",(2,4),SW); | ||
+ | label(scale(ts)*"$+3$",(2,3),NW); | ||
+ | |||
+ | label(scale(ts)*"$\leftarrow$",(5,1),NE); | ||
+ | label(scale(ts)*"$-1$",(5,1),NW); | ||
+ | label(scale(ts)*"$\leftarrow$",(4,1),NE); | ||
+ | label(scale(ts)*"$-2$",(4,1),NW); | ||
+ | label(scale(ts)*"$\leftarrow$",(6,0),NE); | ||
+ | label(scale(ts)*"$-1$",(6,0),NW); | ||
+ | label(scale(ts)*"$\leftarrow$",(5,0),NE); | ||
+ | label(scale(ts)*"$-2$",(5,0),NW); | ||
+ | |||
+ | add(grid(7,7,grey+linewidth(1.25))); //USES OLYMPIAD.ASY | ||
+ | |||
+ | draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2)); | ||
+ | </asy> | ||
+ | From the four numbers that appear in the shaded squares, <math>\boxed{\textbf{(D)}\ 3}</math> of them are prime: <math>19,23,</math> and <math>47.</math> | ||
+ | |||
+ | ~TheMathGuyd | ||
+ | |||
+ | ==Solution 3== | ||
+ | |||
+ | Since the last shaded square is very close to the final number, which is 49, we can easily subtract two instead of having to count all the way there. Then, we can count (:P). | ||
+ | |||
+ | ~AliceDubbleYou | ||
+ | |||
+ | ==Simple, Intuitive Solution by MathTalks_Now== | ||
+ | * *Different Solution not shown before!* | ||
+ | |||
+ | https://studio.youtube.com/video/PMOeiGLkDH0/edit | ||
− | + | ==Video Solution (How to Creatively THINK!!!)== | |
+ | https://youtu.be/7gwhzjySKl0 | ||
− | ~ | + | ~Education the Study of everything |
+ | |||
+ | ==Video Solution by Math-X (Smart and Simple)== | ||
+ | https://youtu.be/Ku_c1YHnLt0?si=cc_Ii2j2pmT6wOuZ&t=412 | ||
+ | |||
+ | ~Math-X | ||
==Video Solution by Magic Square== | ==Video Solution by Magic Square== | ||
https://youtu.be/-N46BeEKaCQ?t=5392 | https://youtu.be/-N46BeEKaCQ?t=5392 | ||
+ | |||
+ | ==Video Solution by SpreadTheMathLove== | ||
+ | https://www.youtube.com/watch?v=EcrktBc8zrM | ||
+ | ==Video Solution by Interstigation== | ||
+ | https://youtu.be/DBqko2xATxs&t=233 | ||
+ | |||
+ | ==Video Solution by WhyMath== | ||
+ | https://youtu.be/1qwfPJDNYGc | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution by harungurcan== | ||
+ | https://www.youtube.com/watch?v=35BW7bsm_Cg&t=402s | ||
+ | |||
+ | ~harungurcan | ||
+ | |||
+ | ==Video Solution by Dr. David== | ||
+ | https://youtu.be/9ynYd5V0r88 | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2023|num-b=3|num-a=5}} | {{AMC8 box|year=2023|num-b=3|num-a=5}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 20:09, 3 November 2024
Contents
[hide]- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Simple, Intuitive Solution by MathTalks_Now
- 6 Video Solution (How to Creatively THINK!!!)
- 7 Video Solution by Math-X (Smart and Simple)
- 8 Video Solution by Magic Square
- 9 Video Solution by SpreadTheMathLove
- 10 Video Solution by Interstigation
- 11 Video Solution by WhyMath
- 12 Video Solution by harungurcan
- 13 Video Solution by Dr. David
- 14 See Also
Problem
The numbers from to are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number How many of these four numbers are prime?
Solution 1
We fill out the grid, as shown below: From the four numbers that appear in the shaded squares, of them are prime: and
~MathFun1000, MRENTHUSIASM
Solution 2
Note that given time constraint, it's better to only count from perfect squares (in pink), as shown below: From the four numbers that appear in the shaded squares, of them are prime: and
~TheMathGuyd
Solution 3
Since the last shaded square is very close to the final number, which is 49, we can easily subtract two instead of having to count all the way there. Then, we can count (:P).
~AliceDubbleYou
Simple, Intuitive Solution by MathTalks_Now
- *Different Solution not shown before!*
https://studio.youtube.com/video/PMOeiGLkDH0/edit
Video Solution (How to Creatively THINK!!!)
~Education the Study of everything
Video Solution by Math-X (Smart and Simple)
https://youtu.be/Ku_c1YHnLt0?si=cc_Ii2j2pmT6wOuZ&t=412
~Math-X
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=5392
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=EcrktBc8zrM
Video Solution by Interstigation
https://youtu.be/DBqko2xATxs&t=233
Video Solution by WhyMath
~savannahsolver
Video Solution by harungurcan
https://www.youtube.com/watch?v=35BW7bsm_Cg&t=402s
~harungurcan
Video Solution by Dr. David
See Also
2023 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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.