<ol>
<li>In a game there are two kinds of tokens: those worth 3 points each and those worth 5 points each. Ken's tokens are worth a total of 585 points. What is the greatest number of tokens Ken could have?</li>
</ol>
<p>Answer 195</p>
<p>2.
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(@=chairs)
Kayula and Paula are to be assigned seats from the arrangement of chairs shown above. If Kayula cannot sit in the same rowbeither vertically or horizontally, how many different seating assignments are possible for Kayula and Paula? </p>
<p>Let T be the number of tokens worth 3 points, and F the number worth 5. So: 3T + 5F = 585. You want to maximize T + F. That will happen when F is as small as possible, since each F token takes up “more” of the available points than the T token. So try F = 0. That leads to an integer solution for T – i.e. T = 585/3 = 195.</p>
<p>Consider a variant of this problem to test your understanding. Suppose the number of points is 595 instead of 585. So 3T + 5F = 595. Try first F = 0. You get T = 198 1/3. This can’t be the answer since both T and F must be integers. Try F = 1. Now you get T = 196 2/3. No go. So try F = 2. Now T = 195. That’s an integer and now T + F = 197.</p>
<p>How about the case when the number of points is 621?</p>
<p>Sit Kayula. How many distinct seats are possible? Well any of the 12. He sits in (Row, Column), where Row can be be 1, 2, or 3 and Column can be 1, 2, 3 or 4.After you sit Kayula how many of the remaining seats are available to Paula? Easiest is to first find how many seats are excluded. All the seats that remain in the row where Kayula sits (i.e. 2) and all the seats that remain in the column where Kayula sits (i.e 3) are excluded. Also of course the seat where Kayula sits is excluded. So 2 + 3 + 1 = 6 are excluded, leaving 6.</p>
<p>So the answer is 12 (possible seats for Kayula) x 6 (possible seats for Paula) or 72.</p>
<p>Try the case with 4 rows and 5 columns, with the same assignment rules as above.</p>
<p>And for a challenge try the case of 4 rows and 5 columns and three assignments – i.e. Kayula, Paula and Stacy. Here Paula cannot sit in the same Row or Column as Kayula, and Stacy cannot sit in the same Row or Column as either Kayula or Paula.</p>
<p>so, in the case of 162, the answer is 207 because T=207, an integer when S=0 and if there 4 rows and 5 columns with three assignments the answer is 20(possible seats for Kayula) x 12 (possible seats for Paula) x 6 (possible seats for Stacy) = 1440 right?</p>
<p>i will give you hints as opposed to a solution because it is best that you struggle with this problem on your own.</p>
<p>1) write a relationship between the three internal angles of the triangle.
2) this will give you a relationship between x and y.
3) use the inequality x>55, and replace x in terms of y.
4) this will give you the range of values of y.</p>
<p>As part of my continuing goal to be the laziest, least insightful kid who still gets the problem right:</p>
<p>They say x>55. I say try x = 56. So 2x = 112. 56 + 112 = 168. 180 - 168 = 12. So 12 is one possible value for y. </p>
<p>They aren’t going to give me more points for doing this algebraically or for generating more solutions. One possible value is what they asked for, one is what I give them :)</p>
<p>PCK, there is an even lazier kid who gets the problem right.</p>
<p>He starts with the simplistic 3x = 165 and notices the 15 left over for y. Since 55 is too small, he has to reduce y by 3 units at a time. Hence 12 9 6 and 3 as answers. </p>
<p>The laziest gets ALL the answers at once and stays away from hard math involving 112 and 56. :)</p>