Difference between revisions of "2006 AMC 8 Problems/Problem 22"
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==Problem== | ==Problem== | ||
| − | Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell | + | Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell? |
| + | <asy> | ||
| + | path cell=((0,0)--(1,0)--(1,1)--(0,1)--cycle); | ||
| + | path sw=((0,0)--(1,sqrt(3))); | ||
| + | path se=((5,0)--(4,sqrt(3))); | ||
| + | draw(cell, linewidth(1)); | ||
| + | draw(shift(2,0)*cell, linewidth(1)); | ||
| + | draw(shift(4,0)*cell, linewidth(1)); | ||
| + | draw(shift(1,3)*cell, linewidth(1)); | ||
| + | draw(shift(3,3)*cell, linewidth(1)); | ||
| + | draw(shift(2,6)*cell, linewidth(1)); | ||
| + | draw(shift(0.45,1.125)*sw, EndArrow); | ||
| + | draw(shift(2.45,1.125)*sw, EndArrow); | ||
| + | draw(shift(1.45,4.125)*sw, EndArrow); | ||
| + | draw(shift(-0.45,1.125)*se, EndArrow); | ||
| + | draw(shift(-2.45,1.125)*se, EndArrow); | ||
| + | draw(shift(-1.45,4.125)*se, EndArrow); | ||
| + | label("$+$", (1.5,1.5)); | ||
| + | label("$+$", (3.5,1.5)); | ||
| + | label("$+$", (2.5,4.5));</asy> | ||
| + | |||
| + | <math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 35 </math> | ||
==Solution== | ==Solution== | ||
| − | + | If the lower cells contain <math>A, B</math> and <math>C</math>, then the second row will contain | |
| − | A + B and B + C, and the top cell will contain A + 2B + C. To obtain the | + | <math>A + B</math> and <math>B + C</math>, and the top cell will contain <math>A + 2B + C</math>. To obtain the |
| − | smallest sum, place 1 in the center cell and 2 and 3 in the outer ones. The top | + | smallest sum, place <math>1</math> in the center cell and <math>2</math> and <math>3</math> in the outer ones. The top |
| − | number will be 7. For the largest sum, place 9 in the center cell and 7 and 8 in | + | number will be <math>7</math>. For the largest sum, place <math>9</math> in the center cell and <math>7</math> and <math>8</math> in |
| − | the outer ones. This top number will be 33. The difference is 33 - 7 = 26. | + | the outer ones. This top number will be <math>33</math>. The difference is <math>33 - 7 = \boxed{\textbf{(D)}\ 26 |
| + | }</math>. | ||
| + | |||
| + | |||
| + | ==Video Solution== | ||
| + | |||
| + | https://www.youtube.com/watch?v=_FAL-t93VIk ~David | ||
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | https://youtu.be/fSl8DBAk-64 | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC8 box|year=2006|n=II|num-b=21|num-a=23}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 17:10, 8 November 2024
Problem
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?
![[asy] path cell=((0,0)--(1,0)--(1,1)--(0,1)--cycle); path sw=((0,0)--(1,sqrt(3))); path se=((5,0)--(4,sqrt(3))); draw(cell, linewidth(1)); draw(shift(2,0)*cell, linewidth(1)); draw(shift(4,0)*cell, linewidth(1)); draw(shift(1,3)*cell, linewidth(1)); draw(shift(3,3)*cell, linewidth(1)); draw(shift(2,6)*cell, linewidth(1)); draw(shift(0.45,1.125)*sw, EndArrow); draw(shift(2.45,1.125)*sw, EndArrow); draw(shift(1.45,4.125)*sw, EndArrow); draw(shift(-0.45,1.125)*se, EndArrow); draw(shift(-2.45,1.125)*se, EndArrow); draw(shift(-1.45,4.125)*se, EndArrow); label("$+$", (1.5,1.5)); label("$+$", (3.5,1.5)); label("$+$", (2.5,4.5));[/asy]](http://latex.artofproblemsolving.com/5/7/0/570b0c91c3fb3ed033703f1831719e94604f7ae9.png)
Solution
If the lower cells contain 
and 
, then the second row will contain
and 
, and the top cell will contain 
. To obtain the
smallest sum, place 
in the center cell and 
and 
in the outer ones. The top
number will be 
. For the largest sum, place 
in the center cell and and 
in
the outer ones. This top number will be 
. The difference is 
.
Video Solution
https://www.youtube.com/watch?v=_FAL-t93VIk ~David
Video Solution by WhyMath
See Also
| 2006 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
