Difference between revisions of "1990 AJHSME Problems/Problem 11"
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==Problem== | ==Problem== | ||
| − | The numbers on the faces of this cube are consecutive whole numbers. The | + | The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is |
<asy> | <asy> | ||
| Line 16: | Line 16: | ||
The only possibilities for the numbers are <math>11,12,13,14,15,16</math> and <math>10,11,12,13,14,15</math>. | The only possibilities for the numbers are <math>11,12,13,14,15,16</math> and <math>10,11,12,13,14,15</math>. | ||
| − | In the second case, the common sum would be <math>(10+11+12+13+14+15)/ | + | In the second case, the common sum would be <math>(10+11+12+13+14+15)/3=25</math>, so <math>11</math> must be paired with <math>14</math>, which |
it isn't. | it isn't. | ||
| Line 25: | Line 25: | ||
{{AJHSME box|year=1990|num-b=10|num-a=12}} | {{AJHSME box|year=1990|num-b=10|num-a=12}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 14:42, 23 May 2021
Problem
The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is
![[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); label("$15$",(1.5,1.2),N); label("$11$",(4,2.3),N); label("$14$",(2.5,3.7),N); [/asy]](http://latex.artofproblemsolving.com/1/b/1/1b1a85e70cc87bd44b6976f6eb76b21a969c7729.png)
Solution
The only possibilities for the numbers are 
and 
.
In the second case, the common sum would be 
, so 
must be paired with 
, which
it isn't.
Thus, the only possibility is the first case and the sum of the six numbers is 
.
See Also
| 1990 AJHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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. 
