Difference between revisions of "Sticky Gum Problem POW"

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<big>'''[[IAG 1H]] [[POW]] <nowiki>#</nowiki> 4: A Sticky Gum Problem'''</big>
 
<big>'''[[IAG 1H]] [[POW]] <nowiki>#</nowiki> 4: A Sticky Gum Problem'''</big>
 
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{{IMP Takedown}}
{{Word|Image:A_Sticky_Gum_Problem.doc}}
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{{PDF|Image:A_Sticky_Gum_Problem.pdf}}
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==Problem Statement==
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Not necessary to do.
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==Process and answers to problems 1,2 & 3==
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1. (2 colors, 2 people) Ms. Hernandez can only spend 3 cents, because on her first cent, she can get a white or a red gumball.  On her second turn, she can also only get a red or  white gumball.  Now she can only have 4 combinations:  Red and White Gumballs;  Red and Red Gumballs;  White and White Gumballs;  White and Red Gumballs.  With 2 of these possible combinations, she already has her goal of having 2 of the same color gumballs.  On her third try, she gets another red or white gumball.  Whatever the color, she already has one of them, which makes 2 of the same color.
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[[Image:Sticky Gum Problem Diagram 1.png]]
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2.  (3 colors, 2 people) Ms. Hernandez now finds a machine that has 3 colors in it.  The most that she will need to spend to get 2 of the same color is 4 cents.  To find all of the possible strategies, look at the chart:
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[[Image:Sticky Gum Problem Diagram 2.png]]
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3. (3 colors, 3 people)  It will take 7 cents to get 3 of the same color, as there are 108 possible combinations.  Here is a chart showing the 1st third of them (if the first color is red)
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[[Image:Sticky Gum Problem Diagram 3.png]]
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I will now make a chart showing my findings so far.
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{|border="2" cellspacing="0" cellpadding="4"
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|align = "center"|<u>'''Colors'''</u>
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|align = "center"|<u>'''Kids'''</u>
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|align = "center"|<u>'''Max Spend'''</u>
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|align = "center"|<u>'''Combos'''</u>
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|-
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|align = "center"|2
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|align = "center"|2
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|align = "center"|3 cents
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|align = "center"|6
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|-
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|align = "center"|3
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|align = "center"|2
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|align = "center"|4 cents
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|align = "center"|33
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|-
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|align = "center"|3
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|align = "center"|3
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|align = "center"|7 cents
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|align = "center"|108
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|-
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|}
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Now I will make up some problems to help fill in the chart some more.
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4. (2 colors, 3 kids)  This chart shows the first half of needing 3 of the same color, with only 2 colors.  It will take 5 cents and there are 16 combinations.
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[[Image:Sticky Gum Problem Diagram 4.png]]
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5. (2 colors, 4 kids)  This shows the first half of getting 4 of 1 color and having 2 colors.  You need 7 cents, and there are 68 combos.
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[[Image:Sticky Gum Problem Diagram 5.png]]
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6. (4 colors, 2 kids) This shows the first quarter of the chart, when you need 2 of the same, and there are 4 colors.  You need 5 turns to get 4 of the same, and there are 200 combinations.
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[[Image:Sticky Gum Problem Diagram 6.png]]
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==Solution (Ultimate Goal)==
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Let me make another chart.  I have included 1 color and 1 kid for comparison.
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{|border="2" cellspacing="0" cellpadding="4"
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|align = "center"|<u>'''Colors'''</u>
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|align = "center"|<u>'''Kids'''</u>
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|align = "center"|<u>'''Max Spend'''</u>
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|align = "center"|<u>'''Combos'''</u>
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|-
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|align = "center"|<font color="#999999">1</font>
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|align = "center"|<font color="#999999">1</font>
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|align = "center"|<font color="#999999">1</font>
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|align = "center"|<font color="#999999">1</font>
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|-
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|align = "center"|<font color="#999999">1</font>
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|align = "center"|<font color="#999999">2</font>
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|align = "center"|<font color="#999999">2</font>
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|align = "center"|<font color="#999999">1</font>
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|-
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|align = "center"|<font color="#999999">2</font>
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|align = "center"|<font color="#999999">1</font>
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|align = "center"|<font color="#999999">1</font>
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|align = "center"|<font color="#999999">2</font>
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|-
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|align = "center"|2
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|align = "center"|2
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|align = "center"|3 cents
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|align = "center"|6
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|-
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|align = "center"|2
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|align = "center"|3
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|align = "center"|5
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|align = "center"|16
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|-
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|align = "center"|2
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|align = "center"|4
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|align = "center"|7
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|align = "center"|68
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|-
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|align = "center"|3
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|align = "center"|2
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|align = "center"|4 cents
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|align = "center"|33
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|-
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|align = "center"|3
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|align = "center"|3
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|align = "center"|7 cents
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|align = "center"|108
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|-
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|align = "center"|4
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|align = "center"|2
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|align = "center"|5
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|align = "center"|200
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|-
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|}
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Overall, I have found that number of colors is ultimately responsible for combinations, but the number of kids is ultimately responsible for the maximum, you spend.  Here is something interesting:
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{|border="2" cellspacing="0" cellpadding="4"
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|align = "center"|<u>'''Colors'''</u>
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|align = "center"|<u>'''Kids'''</u>
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|align = "center"|<u>'''Max Spend'''</u>
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|-
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|align = "center"|2
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|align = "center" rowspan = "3"|<big>2</big>
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|align = "center"|3
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|-
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|align = "center"|3
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|align = "center"|4
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|-
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|align = "center"|4
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|align = "center"|5
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|-
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|}
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When you have 2 kids max spend is equal to number of colors plus 1.  What about having 3 kids:
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{|border="2" cellspacing="0" cellpadding="4"
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|align = "center"|<u>'''Colors'''</u>
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|align = "center"|<u>'''Kids'''</u>
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|align = "center"|<u>'''Max Spend'''</u>
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|-
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|align = "center"|2
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|align = "center" rowspan = "2"|<big>3</big>
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|align = "center"|5
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|-
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|align = "center"|3
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|align = "center"|4
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|-
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|}
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This chart shows so far that when you add a color, the max that you spend, goes down.
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When you start adding a color, the number of cents goes down; when you then keep adding people, the number of cents goes up. 
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I have also found that this works:
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<big><nowiki>[</nowiki>(<nowiki>#</nowiki> of colors) <nowiki>*</nowiki> (<nowiki>#</nowiki> of kids)<nowiki>]</nowiki> - <nowiki>[</nowiki>(<nowiki>#</nowiki> of colors) – 1<nowiki>]</nowiki></big>
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==Extension==
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Not necessary to do.
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==Evaluation==
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Not necessary to do.
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[[Category:IAG 1H]]
 
[[Category:IAG 1H]]
 
[[Category:POW]]
 
[[Category:POW]]

Latest revision as of 16:19, 11 December 2008

IAG 1H POW # 4: A Sticky Gum Problem

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I was asked to take down POW solutions. Remember it's best to think about how to solve the problems on your own.