Sticky Gum Problem POW - Revision history

ThePlaz: imp takedown

2008-12-11T16:19:21Z

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ThePlaz: Reverted edits by Aznboy206 (Talk); changed back to last version by ThePlaz

2007-11-11T16:03:31Z

Reverted edits by Aznboy206 (Talk); changed back to last version by ThePlaz

Aznboy206: /* Solution (Ultimate Goal) */

2007-11-06T13:50:08Z

‎Solution (Ultimate Goal)

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2007-06-22T01:38:30Z

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2007-06-22T01:25:45Z

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2007-06-22T01:04:24Z

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<big>'''[[IAG 1H]] POW <nowiki>#</nowiki> 4: A Sticky Gum Problem'''</big>

{{Word|Image:A_Sticky_Gum_Problem.doc}}
{{PDF|Image:A_Sticky_Gum_Problem.pdf}}

''1. Problem Statement: ''Not necessary to do.

''2. Process and answers to problems 1,2 & 3: ''
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.

[[Image:Sticky Gum Problem Diagram 1.png]]

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:

[[Image:Sticky Gum Problem Diagram 2.png]]

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)

[[Image:Sticky Gum Problem Diagram 3.png]]

I will now make a chart showing my findings so far.

{|border="2" cellspacing="0" cellpadding="4"
|align = "center"|'''Colors'''
|align = "center"|'''Kids'''
|align = "center"|'''Max Spend'''
|align = "center"|'''Combos'''
|-
|align = "center"|2
|align = "center"|2
|align = "center"|3 cents
|align = "center"|6
|-
|align = "center"|3
|align = "center"|2
|align = "center"|4 cents
|align = "center"|33
|-
|align = "center"|3
|align = "center"|3
|align = "center"|7 cents
|align = "center"|108
|-
|}

Now I will make up some problems to help fill in the chart some more.

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.

[[Image:Sticky Gum Problem Diagram 4.png]]

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.

[[Image:Sticky Gum Problem Diagram 5.png]]

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.

[[Image:Sticky Gum Problem Diagram 6.png]]

''3. Solution: ''3. Solution (Ultimate Goal):

Let me make another chart. I have included 1 color and 1 kid for comparison.

{|border="2" cellspacing="0" cellpadding="4"
|align = "center"|'''Colors'''
|align = "center"|'''Kids'''
|align = "center"|'''Max Spend'''
|align = "center"|'''Combos'''
|-
|align = "center"|1
|align = "center"|1
|align = "center"|1
|align = "center"|1
|-
|align = "center"|1
|align = "center"|2
|align = "center"|2
|align = "center"|1
|-
|align = "center"|2
|align = "center"|1
|align = "center"|1
|align = "center"|2
|-
|align = "center"|2
|align = "center"|2
|align = "center"|3 cents
|align = "center"|6
|-
|align = "center"|2
|align = "center"|3
|align = "center"|5
|align = "center"|16
|-
|align = "center"|2
|align = "center"|4
|align = "center"|7
|align = "center"|68
|-
|align = "center"|3
|align = "center"|2
|align = "center"|4 cents
|align = "center"|33
|-
|align = "center"|3
|align = "center"|3
|align = "center"|7 cents
|align = "center"|108
|-
|align = "center"|4
|align = "center"|2
|align = "center"|5
|align = "center"|200
|-
|}

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:

{|border="2" cellspacing="0" cellpadding="4"
|align = "center"|'''Colors'''
|align = "center"|'''Kids'''
|align = "center"|'''Max Spend'''
|-
|align = "center"|2
|align = "center" rowspan = "3"|<big>2</big>
|align = "center"|3
|-
|align = "center"|3
|align = "center"|4
|-
|align = "center"|4
|align = "center"|5
|-
|}

When you have 2 kids max spend is equal to number of colors plus 1. What about having 3 kids:

{|border="2" cellspacing="0" cellpadding="4"
|align = "center"|'''Colors'''
|align = "center"|'''Kids'''
|align = "center"|'''Max Spend'''
|-
|align = "center"|2
|align = "center" rowspan = "2"|<big>3</big>
|align = "center"|5
|-
|align = "center"|3
|align = "center"|4
|-
|}

This chart shows so far that when you add a color, the max that you spend, goes down.

When you start adding a color, the number of cents goes down; when you then keep adding people, the number of cents goes up.

I have also found that this works:
<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>

''4. Extension: ''Not necessary to do.

''5. Evaluation: ''Not necessary to do.
[[Category:IAG 1H]]
[[Category:POW]]

@@ Line 149: / Line 149: @@
 |-
 |align = "center"|3
-|align = "center"|7
+|align = "center"|4
 |-
 |}
@@ Line 158: / Line 158: @@
 I have also found that this works:
-<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>
+<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>
 ==Extension==

@@ Line 1: / Line 1: @@
-<big>'''[[IAG 1H]] POW <nowiki>#</nowiki> 4: A Sticky Gum Problem'''</big>
+<big>'''[[IAG 1H]] [[POW]] <nowiki>#</nowiki> 4: A Sticky Gum Problem'''</big>
 {{Word|Image:A_Sticky_Gum_Problem.doc}}
 {{PDF|Image:A_Sticky_Gum_Problem.pdf}}
-''1. Problem Statement: ''Not necessary to do.
+==Problem Statement==
-''2. Process and answers to problems 1,2 & 3: ''
+==Process and answers to problems 1,2 & 3==
 . (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.
@@ Line 58: / Line 59: @@
 [[Image:Sticky Gum Problem Diagram 6.png]]
-''3. Solution (Ultimate Goal):  ''
+==Solution (Ultimate Goal)==
 Let me make another chart.  I have included 1 color and 1 kid for comparison.
@@ Line 159: / Line 160: @@
 <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>
-''4. Extension:  ''Not necessary to do.
+==Extension==
 [[Category:IAG 1H]]
 [[Category:POW]]