Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1993 AHSME Problems/Problem 22"

m (The picture was incorrect and unhelpful to solve the problem.)
(Fixing an asymptote diagram.)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
 +
 +
<asy>
 +
size((400));
 +
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0), linewidth(1));
 +
draw((5,0)--(10,0)--(15,0)--(20,0)--(20,5)--(15,5)--(10,5)--(5,5)--(6,7)--(11,7)--(16,7)--(21,7)--(21,2)--(20,0), linewidth(1));
 +
draw((10,0)--(10,5)--(11,7), linewidth(1));
 +
draw((15,0)--(15,5)--(16,7), linewidth(1));
 +
draw((20,0)--(20,5)--(21,7), linewidth(1));
 +
draw((0,5)--(1,7)--(6,7), linewidth(1));
 +
draw((3.5,7)--(4.5,9)--(9.5,9)--(14.5,9)--(19.5,9)--(18.5,7)--(19.5,9)--(19.5,7), linewidth(1));
 +
draw((8.5,7)--(9.5,9), linewidth(1));
 +
draw((13.5,7)--(14.5,9), linewidth(1));
 +
draw((7,9)--(8,11)--(13,11)--(18,11)--(17,9)--(18,11)--(18,9), linewidth(1));
 +
draw((12,9)--(13,11), linewidth(1));
 +
draw((10.5,11)--(11.5,13)--(16.5,13)--(16.5,11)--(16.5,13)--(15.5,11), linewidth(1));
 +
draw((25,0)--(30,0)--(30,5)--(25,5)--(25,0), dashed);
 +
draw((30,0)--(35,0)--(40,0)--(45,0)--(45,5)--(40,5)--(35,5)--(30,5)--(31,7)--(36,7)--(41,7)--(46,7)--(46,2)--(45,0), dashed);
 +
draw((35,0)--(35,5)--(36,7), dashed);
 +
draw((40,0)--(40,5)--(41,7), dashed);
 +
draw((45,0)--(45,5)--(46,7), dashed);
 +
draw((25,5)--(26,7)--(31,7), dashed);
 +
draw((28.5,7)--(29.5,9)--(34.5,9)--(39.5,9)--(44.5,9)--(43.5,7)--(44.5,9)--(44.5,7), dashed);
 +
draw((33.5,7)--(34.5,9), dashed);
 +
draw((38.5,7)--(39.5,9), dashed);
 +
draw((32,9)--(33,11)--(38,11)--(43,11)--(42,9)--(43,11)--(43,9), dashed);
 +
draw((37,9)--(38,11), dashed);
 +
draw((35.5,11)--(36.5,13)--(41.5,13)--(41.5,11)--(41.5,13)--(40.5,11), dashed);
 +
draw((50,0)--(55,0)--(55,5)--(50,5)--(50,0), dashed);
 +
draw((55,0)--(60,0)--(65,0)--(70,0)--(70,5)--(65,5)--(60,5)--(55,5)--(56,7)--(61,7)--(66,7)--(71,7)--(71,2)--(70,0), dashed);
 +
draw((60,0)--(60,5)--(61,7), dashed);
 +
draw((65,0)--(65,5)--(66,7), dashed);
 +
draw((70,0)--(70,5)--(71,7), dashed);
 +
draw((50,5)--(51,7)--(56,7), dashed);
 +
draw((53.5,7)--(54.5,9)--(59.5,9)--(64.5,9)--(69.5,9)--(68.5,7)--(69.5,9)--(69.5,7), dashed);
 +
draw((58.5,7)--(59.5,9), dashed);
 +
draw((63.5,7)--(64.5,9), dashed);
 +
draw((57,9)--(58,11)--(63,11)--(68,11)--(67,9)--(68,11)--(68,9), dashed);
 +
draw((62,9)--(63,11), dashed);
 +
draw((60.5,11)--(61.5,13)--(66.5,13)--(66.5,11)--(66.5,13)--(65.5,11), dashed);
 +
draw((75,0)--(80,0)--(80,5)--(75,5)--(75,0), dashed);
 +
draw((80,0)--(85,0)--(90,0)--(95,0)--(95,5)--(90,5)--(85,5)--(80,5)--(81,7)--(86,7)--(91,7)--(96,7)--(96,2)--(95,0), dashed);
 +
draw((85,0)--(85,5)--(86,7), dashed);
 +
draw((90,0)--(90,5)--(91,7), dashed);
 +
draw((95,0)--(95,5)--(96,7), dashed);
 +
draw((75,5)--(76,7)--(81,7), dashed);
 +
draw((78.5,7)--(79.5,9)--(84.5,9)--(89.5,9)--(94.5,9)--(93.5,7)--(94.5,9)--(94.5,7), dashed);
 +
draw((83.5,7)--(84.5,9), dashed);
 +
draw((88.5,7)--(89.5,9), dashed);
 +
draw((82,9)--(83,11)--(88,11)--(93,11)--(92,9)--(93,11)--(93,9), dashed);
 +
draw((87,9)--(88,11), dashed);
 +
draw((85.5,11)--(86.5,13)--(91.5,13)--(91.5,11)--(91.5,13)--(90.5,11), dashed);
 +
draw((28,6)--(33,6)--(38,6)--(43,6)--(43,11)--(38,11)--(33,11)--(28,11)--(28,6), linewidth(1));
 +
draw((28,11)--(29,13)--(34,13)--(39,13)--(44,13)--(43,11)--(44,13)--(44,8)--(43,6), linewidth(1));
 +
draw((33,6)--(33,11)--(34,13)--(39,13)--(38,11)--(38,6), linewidth(1));
 +
draw((31,13)--(32,15)--(37,15)--(36,13)--(37,15)--(42,15)--(41,13)--(42,15)--(42,13), linewidth(1));
 +
draw((34.5,15)--(35.5,17)--(40.5,17)--(39.5,15)--(40.5,17)--(40.5,15), linewidth(1));
 +
draw((53,6)--(58,6)--(63,6)--(68,6)--(68,11)--(63,11)--(58,11)--(53,11)--(53,6), dashed);
 +
draw((53,11)--(54,13)--(59,13)--(64,13)--(69,13)--(68,11)--(69,13)--(69,8)--(68,6), dashed);
 +
draw((58,6)--(58,11)--(59,13)--(64,13)--(63,11)--(63,6), dashed);
 +
draw((56,13)--(57,15)--(62,15)--(61,13)--(62,15)--(67,15)--(66,13)--(67,15)--(67,13), dashed);
 +
draw((59.5,15)--(60.5,17)--(65.5,17)--(64.5,15)--(65.5,17)--(65.5,15), dashed);
 +
draw((78,6)--(83,6)--(88,6)--(93,6)--(93,11)--(88,11)--(83,11)--(78,11)--(78,6), dashed);
 +
draw((78,11)--(79,13)--(84,13)--(89,13)--(94,13)--(93,11)--(94,13)--(94,8)--(93,6), dashed);
 +
draw((83,6)--(83,11)--(84,13)--(89,13)--(88,11)--(88,6), dashed);
 +
draw((81,13)--(82,15)--(87,15)--(86,13)--(87,15)--(92,15)--(91,13)--(92,15)--(92,13), dashed);
 +
draw((84.5,15)--(85.5,17)--(90.5,17)--(89.5,15)--(90.5,17)--(90.5,15), dashed);
 +
draw((56,12)--(61,12)--(66,12)--(66,17)--(61,17)--(56,17)--(56,12), linewidth(1));
 +
draw((61,12)--(61,17)--(62,19)--(57,19)--(56,17)--(57,19)--(67,19)--(66,17)--(67,19)--(67,14)--(66,12), linewidth(1));
 +
draw((59.5,19)--(60.5,21)--(65.5,21)--(64.5,19)--(65.5,21)--(65.5,19), linewidth(1));
 +
draw((81,12)--(86,12)--(91,12)--(91,17)--(86,17)--(81,17)--(81,12), dashed);
 +
draw((86,12)--(86,17)--(87,19)--(82,19)--(81,17)--(82,19)--(92,19)--(91,17)--(92,19)--(92,14)--(91,12), dashed);
 +
draw((84.5,19)--(85.5,21)--(90.5,21)--(89.5,19)--(90.5,21)--(90.5,19), dashed);
 +
draw((84,18)--(89,18)--(89,23)--(84,23)--(84,18)--(84,23)--(85,25)--(90,25)--(89,23)--(90,25)--(90,20)--(89,18), linewidth(1));</asy>
 +
 
Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.
 
Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.
  
Line 9: Line 83:
  
 
== Solution ==
 
== Solution ==
−
<math>\fbox{C}</math>
+
The value assigned at the top is the weighted sum of the values in the cubes of a given level.  The weight for a given cube is the sum of the weights of the blocks which touch it above, since the number in a cube will get incorporated into the overall sum via each of those blocks, according to the weight of each block.  Thus, if we think about how the weights propagate down the block pyramid, its sort of like a three dimensional Pascal's triangle.
 +
 
 +
The first layer is <math>1</math>
 +
 
 +
The second layer is
 +
 
 +
<math>| \, \, 1 \\ |1 \, 1</math>
 +
 
 +
The third layer is
 +
 
 +
<math>| \,\,\,\, 1 \\| \,\, 2 \, 2 \\| 1 \,2 \, 1</math>
 +
 
 +
The fourth layer is
 +
 
 +
<math>| \,\,\,\,\,\, 1 \\| \,\,\,\, 3 \, 3 \\| \,\, 3 \,6 \, 3 \\| 1\, 3\, 3\, 1</math>
 +
 
 +
The top level sum is minimized if we associate the block with weight 6 with the smallest number 1, the blocks with weight 1 with the largest numbers 8, 9 and 10, and the rest of the blocks with weight 3 with the rest of the numbers.  The sum is <math>6\cdot 1 + 3 \cdot (2+3+4+5+6+7 ) + 1\cdot (8+9+10) = 114</math> so the answer is  <math>\fbox{C}</math>
  
 
== See also ==
 
== See also ==

Latest revision as of 08:19, 27 June 2021

Problem

[asy] size((400)); draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0), linewidth(1)); draw((5,0)--(10,0)--(15,0)--(20,0)--(20,5)--(15,5)--(10,5)--(5,5)--(6,7)--(11,7)--(16,7)--(21,7)--(21,2)--(20,0), linewidth(1)); draw((10,0)--(10,5)--(11,7), linewidth(1)); draw((15,0)--(15,5)--(16,7), linewidth(1)); draw((20,0)--(20,5)--(21,7), linewidth(1)); draw((0,5)--(1,7)--(6,7), linewidth(1)); draw((3.5,7)--(4.5,9)--(9.5,9)--(14.5,9)--(19.5,9)--(18.5,7)--(19.5,9)--(19.5,7), linewidth(1)); draw((8.5,7)--(9.5,9), linewidth(1)); draw((13.5,7)--(14.5,9), linewidth(1)); draw((7,9)--(8,11)--(13,11)--(18,11)--(17,9)--(18,11)--(18,9), linewidth(1)); draw((12,9)--(13,11), linewidth(1)); draw((10.5,11)--(11.5,13)--(16.5,13)--(16.5,11)--(16.5,13)--(15.5,11), linewidth(1)); draw((25,0)--(30,0)--(30,5)--(25,5)--(25,0), dashed); draw((30,0)--(35,0)--(40,0)--(45,0)--(45,5)--(40,5)--(35,5)--(30,5)--(31,7)--(36,7)--(41,7)--(46,7)--(46,2)--(45,0), dashed); draw((35,0)--(35,5)--(36,7), dashed); draw((40,0)--(40,5)--(41,7), dashed); draw((45,0)--(45,5)--(46,7), dashed); draw((25,5)--(26,7)--(31,7), dashed); draw((28.5,7)--(29.5,9)--(34.5,9)--(39.5,9)--(44.5,9)--(43.5,7)--(44.5,9)--(44.5,7), dashed); draw((33.5,7)--(34.5,9), dashed); draw((38.5,7)--(39.5,9), dashed); draw((32,9)--(33,11)--(38,11)--(43,11)--(42,9)--(43,11)--(43,9), dashed); draw((37,9)--(38,11), dashed); draw((35.5,11)--(36.5,13)--(41.5,13)--(41.5,11)--(41.5,13)--(40.5,11), dashed); draw((50,0)--(55,0)--(55,5)--(50,5)--(50,0), dashed); draw((55,0)--(60,0)--(65,0)--(70,0)--(70,5)--(65,5)--(60,5)--(55,5)--(56,7)--(61,7)--(66,7)--(71,7)--(71,2)--(70,0), dashed); draw((60,0)--(60,5)--(61,7), dashed); draw((65,0)--(65,5)--(66,7), dashed); draw((70,0)--(70,5)--(71,7), dashed); draw((50,5)--(51,7)--(56,7), dashed); draw((53.5,7)--(54.5,9)--(59.5,9)--(64.5,9)--(69.5,9)--(68.5,7)--(69.5,9)--(69.5,7), dashed); draw((58.5,7)--(59.5,9), dashed); draw((63.5,7)--(64.5,9), dashed); draw((57,9)--(58,11)--(63,11)--(68,11)--(67,9)--(68,11)--(68,9), dashed); draw((62,9)--(63,11), dashed); draw((60.5,11)--(61.5,13)--(66.5,13)--(66.5,11)--(66.5,13)--(65.5,11), dashed); draw((75,0)--(80,0)--(80,5)--(75,5)--(75,0), dashed); draw((80,0)--(85,0)--(90,0)--(95,0)--(95,5)--(90,5)--(85,5)--(80,5)--(81,7)--(86,7)--(91,7)--(96,7)--(96,2)--(95,0), dashed); draw((85,0)--(85,5)--(86,7), dashed); draw((90,0)--(90,5)--(91,7), dashed); draw((95,0)--(95,5)--(96,7), dashed); draw((75,5)--(76,7)--(81,7), dashed); draw((78.5,7)--(79.5,9)--(84.5,9)--(89.5,9)--(94.5,9)--(93.5,7)--(94.5,9)--(94.5,7), dashed); draw((83.5,7)--(84.5,9), dashed); draw((88.5,7)--(89.5,9), dashed); draw((82,9)--(83,11)--(88,11)--(93,11)--(92,9)--(93,11)--(93,9), dashed); draw((87,9)--(88,11), dashed); draw((85.5,11)--(86.5,13)--(91.5,13)--(91.5,11)--(91.5,13)--(90.5,11), dashed); draw((28,6)--(33,6)--(38,6)--(43,6)--(43,11)--(38,11)--(33,11)--(28,11)--(28,6), linewidth(1)); draw((28,11)--(29,13)--(34,13)--(39,13)--(44,13)--(43,11)--(44,13)--(44,8)--(43,6), linewidth(1)); draw((33,6)--(33,11)--(34,13)--(39,13)--(38,11)--(38,6), linewidth(1)); draw((31,13)--(32,15)--(37,15)--(36,13)--(37,15)--(42,15)--(41,13)--(42,15)--(42,13), linewidth(1)); draw((34.5,15)--(35.5,17)--(40.5,17)--(39.5,15)--(40.5,17)--(40.5,15), linewidth(1)); draw((53,6)--(58,6)--(63,6)--(68,6)--(68,11)--(63,11)--(58,11)--(53,11)--(53,6), dashed); draw((53,11)--(54,13)--(59,13)--(64,13)--(69,13)--(68,11)--(69,13)--(69,8)--(68,6), dashed); draw((58,6)--(58,11)--(59,13)--(64,13)--(63,11)--(63,6), dashed); draw((56,13)--(57,15)--(62,15)--(61,13)--(62,15)--(67,15)--(66,13)--(67,15)--(67,13), dashed); draw((59.5,15)--(60.5,17)--(65.5,17)--(64.5,15)--(65.5,17)--(65.5,15), dashed); draw((78,6)--(83,6)--(88,6)--(93,6)--(93,11)--(88,11)--(83,11)--(78,11)--(78,6), dashed); draw((78,11)--(79,13)--(84,13)--(89,13)--(94,13)--(93,11)--(94,13)--(94,8)--(93,6), dashed); draw((83,6)--(83,11)--(84,13)--(89,13)--(88,11)--(88,6), dashed); draw((81,13)--(82,15)--(87,15)--(86,13)--(87,15)--(92,15)--(91,13)--(92,15)--(92,13), dashed); draw((84.5,15)--(85.5,17)--(90.5,17)--(89.5,15)--(90.5,17)--(90.5,15), dashed); draw((56,12)--(61,12)--(66,12)--(66,17)--(61,17)--(56,17)--(56,12), linewidth(1)); draw((61,12)--(61,17)--(62,19)--(57,19)--(56,17)--(57,19)--(67,19)--(66,17)--(67,19)--(67,14)--(66,12), linewidth(1)); draw((59.5,19)--(60.5,21)--(65.5,21)--(64.5,19)--(65.5,21)--(65.5,19), linewidth(1)); draw((81,12)--(86,12)--(91,12)--(91,17)--(86,17)--(81,17)--(81,12), dashed); draw((86,12)--(86,17)--(87,19)--(82,19)--(81,17)--(82,19)--(92,19)--(91,17)--(92,19)--(92,14)--(91,12), dashed); draw((84.5,19)--(85.5,21)--(90.5,21)--(89.5,19)--(90.5,21)--(90.5,19), dashed); draw((84,18)--(89,18)--(89,23)--(84,23)--(84,18)--(84,23)--(85,25)--(90,25)--(89,23)--(90,25)--(90,20)--(89,18), linewidth(1));[/asy]

Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.

$\text{(A) } 55\quad \text{(B) } 83\quad \text{(C) } 114\quad \text{(D) } 137\quad \text{(E) } 144$

Solution

The value assigned at the top is the weighted sum of the values in the cubes of a given level. The weight for a given cube is the sum of the weights of the blocks which touch it above, since the number in a cube will get incorporated into the overall sum via each of those blocks, according to the weight of each block. Thus, if we think about how the weights propagate down the block pyramid, its sort of like a three dimensional Pascal's triangle.

The first layer is $1$

The second layer is

$| \, \, 1 \\ |1 \, 1$

The third layer is

$| \,\,\,\, 1 \\| \,\, 2 \, 2 \\| 1 \,2 \, 1$

The fourth layer is

$| \,\,\,\,\,\, 1 \\| \,\,\,\, 3 \, 3 \\| \,\, 3 \,6 \, 3 \\| 1\, 3\, 3\, 1$

The top level sum is minimized if we associate the block with weight 6 with the smallest number 1, the blocks with weight 1 with the largest numbers 8, 9 and 10, and the rest of the blocks with weight 3 with the rest of the numbers. The sum is $6\cdot 1 + 3 \cdot (2+3+4+5+6+7 ) + 1\cdot (8+9+10) = 114$ so the answer is $\fbox{C}$

See also

1993 AHSME (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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1993_AHSME_Problems/Problem_22&oldid=156931"