Difference between revisions of "2024 AMC 8 Problems/Problem 7"
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==Problem== | ==Problem== | ||
+ | A <math>3 \times 7</math> rectangle is covered without overlap by 3 shapes of tiles: <math>2 \times 2</math>, <math>1\times4</math>, and <math>1\times1</math>, shown below. What is the minimum possible number of <math>1\times1</math> tiles used? | ||
− | + | [[File:2024-AMC8-q7.png|center]] | |
− | <math>\textbf{(A) } | + | <math>\textbf{(A) } 1\qquad\textbf{(B)} 2\qquad\textbf{(C) } 3\qquad\textbf{(D) } 4\qquad\textbf{(E) } 5</math> |
+ | ==Solution 1== | ||
+ | |||
+ | We can eliminate B, C, and D, because they are not <math>21</math> subtracted by any multiple of <math>4</math>. Finally, we see that there is no way to have A, so the solution is <math>\boxed{\textbf{(E)\ 5}}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | Let <math>x</math> be the number of <math>1x1</math> tiles. There are <math>21</math> squares and each <math>2x2</math> or <math>1x4</math> tile takes up 4 squares, so <math>x \equiv 1 \pmod{4}</math>, so it is either <math>1</math> or <math>5</math>. Color the columns, starting with red, then blue, and alternating colors, ending with a red column. There are <math>12</math> red squares and <math>9</math> blue squares, but each <math>2x2</math> and <math>1x4</math> shape takes up an equal number of blue and red squares, so there must be <math>3</math> more <math>1x1</math> tiles on red squares than on blue squares, which is impossible if there is just one, so the answer is <math>\boxed{\textbf{(E)\ 5}}</math>, which can easily be confirmed to work. | ||
+ | |||
+ | ~arfekete | ||
+ | |||
+ | ==Solution 3== | ||
+ | Suppose there are <math>a</math> different <math>2\times 2</math> tiles, <math>b</math> different <math>4\times 1</math> tiles and <math>c</math> different <math>1\times 1</math> tiles. Since the areas of these tiles must total up to <math>21</math> (area of the whole grid), we have | ||
+ | <cmath>4a + 4b + c = 21.</cmath> | ||
+ | Reducing modulo <math>4</math> gives <math>c\equiv 1\pmod{4}</math>, or <math>c = 1</math> or <math>c = 5</math>. | ||
+ | |||
+ | If <math>c = 1</math>, then <math>a + b = 5</math>. After some testing, there is no valid pair <math>(a, b)</math> that works, so the answer must be <math>\boxed{\textbf{(E)\ 5}}</math>, which can be constructed in many ways. | ||
+ | |||
+ | -Benedict T (countmath1) | ||
+ | |||
+ | ==Video Solution 1 (easy to digest) by Power Solve== | ||
+ | https://youtu.be/16YYti_pDUg?si=KjRhUdCOAx10kgiW&t=59 | ||
+ | |||
+ | ==Video Solution by Math-X (First fully understand the problem!!!)== | ||
+ | https://youtu.be/BaE00H2SHQM?si=eg8T8wApi2j83Ia_&t=1497 | ||
+ | ~Math-X | ||
+ | |||
+ | ==Video Solution by NiuniuMaths (Easy to understand!)== | ||
+ | https://www.youtube.com/watch?v=V-xN8Njd_Lc | ||
− | + | ~NiuniuMaths | |
− | ==Solution | + | ==Video Solution 2 by SpreadTheMathLove== |
+ | https://www.youtube.com/watch?v=L83DxusGkSY | ||
+ | |||
+ | == Video Solution by CosineMethod [🔥Fast and Easy🔥]== | ||
+ | |||
+ | https://www.youtube.com/watch?v=L4ouVVVkFo4 | ||
+ | ==Video Solution by Interstigation== | ||
+ | https://youtu.be/ktzijuZtDas&t=578 | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2024|num-b=6|num-a=8}} | ||
+ | {{MAA Notice}} |
Revision as of 15:55, 20 February 2024
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Video Solution 1 (easy to digest) by Power Solve
- 6 Video Solution by Math-X (First fully understand the problem!!!)
- 7 Video Solution by NiuniuMaths (Easy to understand!)
- 8 Video Solution 2 by SpreadTheMathLove
- 9 Video Solution by CosineMethod [🔥Fast and Easy🔥]
- 10 Video Solution by Interstigation
- 11 See Also
Problem
A rectangle is covered without overlap by 3 shapes of tiles:
,
, and
, shown below. What is the minimum possible number of
tiles used?
Solution 1
We can eliminate B, C, and D, because they are not subtracted by any multiple of
. Finally, we see that there is no way to have A, so the solution is
.
Solution 2
Let be the number of
tiles. There are
squares and each
or
tile takes up 4 squares, so
, so it is either
or
. Color the columns, starting with red, then blue, and alternating colors, ending with a red column. There are
red squares and
blue squares, but each
and
shape takes up an equal number of blue and red squares, so there must be
more
tiles on red squares than on blue squares, which is impossible if there is just one, so the answer is
, which can easily be confirmed to work.
~arfekete
Solution 3
Suppose there are different
tiles,
different
tiles and
different
tiles. Since the areas of these tiles must total up to
(area of the whole grid), we have
Reducing modulo
gives
, or
or
.
If , then
. After some testing, there is no valid pair
that works, so the answer must be
, which can be constructed in many ways.
-Benedict T (countmath1)
Video Solution 1 (easy to digest) by Power Solve
https://youtu.be/16YYti_pDUg?si=KjRhUdCOAx10kgiW&t=59
Video Solution by Math-X (First fully understand the problem!!!)
https://youtu.be/BaE00H2SHQM?si=eg8T8wApi2j83Ia_&t=1497 ~Math-X
Video Solution by NiuniuMaths (Easy to understand!)
https://www.youtube.com/watch?v=V-xN8Njd_Lc
~NiuniuMaths
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=L83DxusGkSY
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://www.youtube.com/watch?v=L4ouVVVkFo4
Video Solution by Interstigation
https://youtu.be/ktzijuZtDas&t=578
See Also
2024 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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.