Difference between revisions of "2013 AMC 8 Problems/Problem 22"
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~Note by Theraccoon: The person who posted this answer did not include their name. Minor edit by ~NXC | ~Note by Theraccoon: The person who posted this answer did not include their name. Minor edit by ~NXC | ||
| − | ==Solution 2 - Common | + | ==Solution 2 - Common Sense== |
With a quick mental calculation, 60 * 30 yields 1800, which is roughly where 4 of our 5 answer choices lie in. However, we can tell that each square would require at least 2 toothpicks that uniquely belong to itself, so the answer would be <math>60\cdot 30 \cdot 2</math> which would be roughly <math>\boxed{\textbf{(E)}\ 3932}</math>. | With a quick mental calculation, 60 * 30 yields 1800, which is roughly where 4 of our 5 answer choices lie in. However, we can tell that each square would require at least 2 toothpicks that uniquely belong to itself, so the answer would be <math>60\cdot 30 \cdot 2</math> which would be roughly <math>\boxed{\textbf{(E)}\ 3932}</math>. | ||
-superplayer24 | -superplayer24 | ||
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==See Also== | ==See Also== | ||
{{AMC8 box|year=2013|num-b=21|num-a=23}} | {{AMC8 box|year=2013|num-b=21|num-a=23}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 15:03, 24 August 2025
Problem
Toothpicks are used to make a grid that is 
toothpicks long and 
toothpicks wide. How many toothpicks are used altogether?
![[asy] picture corner; draw(corner,(5,0)--(35,0)); draw(corner,(0,-5)--(0,-35)); for (int i=0; i<3; ++i){for (int j=0; j>-2; --j){if ((i-j)<3){add(corner,(50i,50j));}}} draw((5,-100)--(45,-100)); draw((155,0)--(185,0),dotted+linewidth(2)); draw((105,-50)--(135,-50),dotted+linewidth(2)); draw((100,-55)--(100,-85),dotted+linewidth(2)); draw((55,-100)--(85,-100),dotted+linewidth(2)); draw((50,-105)--(50,-135),dotted+linewidth(2)); draw((0,-105)--(0,-135),dotted+linewidth(2));[/asy]](http://latex.artofproblemsolving.com/c/6/1/c610668e8c5623b4247005f01744cde6726b65a9.png)
Video Solution
https://youtu.be/nNDdkv_zfOo ~savannahsolver
Solution 1
There are 
vertical columns with a length of toothpicks, and there are 
horizontal rows with a length of toothpicks, because and are the number of intervals. You can verify this by trying a smaller case, i.e. a 
grid of toothpicks, with 
and 
.
Thus, our answer is 
.
~Note by Theraccoon: The person who posted this answer did not include their name. Minor edit by ~NXC
Solution 2 - Common Sense
With a quick mental calculation, 60 * 30 yields 1800, which is roughly where 4 of our 5 answer choices lie in. However, we can tell that each square would require at least 2 toothpicks that uniquely belong to itself, so the answer would be 
which would be roughly 
.
-superplayer24
See Also
| 2013 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
