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Difference between revisions of "2023 AMC 8 Problems/Problem 4"

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==Video Solution (How to Creatively THINK!!!)==
 
==Video Solution (How to Creatively THINK!!!)==

Revision as of 21:15, 29 December 2023

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? [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] $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

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, $\boxed{\textbf{(D)}\ 3}$ of them are prime: $19,23,$ and $47.$

~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, $\boxed{\textbf{(D)}\ 3}$ of them are prime: $19,23,$ and $47.$

~TheMathGuyd


*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

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

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. AMC logo.png

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