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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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| − | {{Word|Image:A_Sticky_Gum_Problem.doc}} | + | |
| − | {{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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| − | 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"|7
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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]] |
IMP Takedown
I was asked to take down POW solutions. Remember it's best to think about how to solve the problems on your own.
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