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Difference between revisions of "1998 AJHSME Problems/Problem 24"
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A rectangular board of 8 columns has squared numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?
 
A rectangular board of 8 columns has squared numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?
  
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[[File:[asy]
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[http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2331780&sid=3057ce3af8558814a3f7473fd1629118#p2331780 picture]
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unitsize(20);
 
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for(int a = 0; a < 10; ++a)
 
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{
 
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draw((0,a)--(8,a));
 
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}
 
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for (int b = 0; b < 9; ++b)
 
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{
 
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draw((b,0)--(b,9));
 
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}
 
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draw((0,0)--(0,-.5));
 
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draw((1,0)--(1,-1.5));
 
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draw((.5,-1)--(1.5,-1));
 
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draw((2,0)--(2,-.5));
 
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draw((4,0)--(4,-.5));
 
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draw((5,0)--(5,-1.5));
 
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draw((4.5,-1)--(5.5,-1));
 
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draw((6,0)--(6,-.5));
 
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draw((8,0)--(8,-.5));
 
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fill((0,8)--(1,8)--(1,9)--(0,9)--cycle,black);
 
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fill((2,8)--(3,8)--(3,9)--(2,9)--cycle,black);
 
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fill((5,8)--(6,8)--(6,9)--(5,9)--cycle,black);
 
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fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black);
 
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fill((6,7)--(7,7)--(7,8)--(6,8)--cycle,black);
 
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label("<math>2</math>",(1.5,8.2),N);
 
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label("<math>4</math>",(3.5,8.2),N);
 
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label("<math>5</math>",(4.5,8.2),N);
 
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label("<math>7</math>",(6.5,8.2),N);
 
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label("<math>8</math>",(7.5,8.2),N);
 
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label("<math>9</math>",(0.5,7.2),N);
 
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label("<math>11</math>",(2.5,7.2),N);
 
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label("<math>12</math>",(3.5,7.2),N);
 
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label("<math>13</math>",(4.5,7.2),N);
 
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label("<math>14</math>",(5.5,7.2),N);
 
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label("<math>16</math>",(7.5,7.2),N);[/asy]]]
 
  
 
<math> \text{(A)}\ 36\qquad\text{(B)}\ 64\qquad\text{(C)}\ 78\qquad\text{(D)}\ 91\qquad\text{(E)}\ 120 </math>
 
<math> \text{(A)}\ 36\qquad\text{(B)}\ 64\qquad\text{(C)}\ 78\qquad\text{(D)}\ 91\qquad\text{(E)}\ 120 </math>

Revision as of 12:21, 27 June 2011

A rectangular board of 8 columns has squared numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?

picture

$\text{(A)}\ 36\qquad\text{(B)}\ 64\qquad\text{(C)}\ 78\qquad\text{(D)}\ 91\qquad\text{(E)}\ 120$

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