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Total Toothpaste (Posted on 2004-02-16) Difficulty: 3 of 5
When it's time to squeeze out toothpaste, one method is to press down on one particular "zone" of the toothpaste tube. This way, half of the toothpaste will squirt in each direction except for the zone at the end of the tube. When toothpaste squirts out the zone at the front of the tube, it comes out onto your toothbrush.

The problem is a person who uses this toothpaste-squeezing method has to go in for a dentist visit tomorrow, and he haven't even opened his tube of toothpaste. He figure if he can get 99% of the toothpaste on his toothbrush and brush with it, he will have shiny teeth. It's late at night and he doesn't want to have to squeeze the toothpaste tube more than neccesary.

A) What would be the best squeezing strategy if the tube had 3 zones? 4 zones?

B) Is there a best strategy that could apply for any number of zones?

Example:
Toothpaste tube-> |4321=
1/4, 1/4, 1/4, 1/4, (  0 OUT) Press 3:
3/8,    , 3/8, 1/4, (  0 OUT) Press 1:
3/8,    , 1/2,    , (1/8 OUT) Press 2:
3/8, 1/4,    , 1/4, (1/8 OUT) Press 4:
   , 5/8,    , 1/4, (1/8 OUT)
(Assume any part of the tube can have any amount of toothpaste in it, you can only push in the middle of a zone, and the person will not get sick from toothpaste intake.)

See The Solution Submitted by Gamer    
Rating: 4.4000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution The 4-zone case laid out | Comment 4 of 5 |
The following shows the zone contents and amount already expelled for the best 4-zone case. Zone 4 is farthest from the opening and zone 1 nearest.

s# zone zone 4 zone 3 zone 2 zone 1 total on brush
1 2 0.25000000 0.37500000 0.00000000 0.37500000 0.00000000
2 4 0.00000000 0.62500000 0.00000000 0.37500000 0.00000000
3 3 0.31250000 0.00000000 0.31250000 0.37500000 0.00000000
4 2 0.31250000 0.15625000 0.00000000 0.53125000 0.00000000
5 1 0.31250000 0.15625000 0.26562500 0.00000000 0.26562500
6 4 0.00000000 0.46875000 0.26562500 0.00000000 0.26562500
7 3 0.23437500 0.00000000 0.50000000 0.00000000 0.26562500
8 2 0.23437500 0.25000000 0.00000000 0.25000000 0.26562500
9 1 0.23437500 0.25000000 0.12500000 0.00000000 0.39062500
10 4 0.00000000 0.48437500 0.12500000 0.00000000 0.39062500
11 3 0.24218750 0.00000000 0.36718750 0.00000000 0.39062500
12 2 0.24218750 0.18359375 0.00000000 0.18359375 0.39062500
13 1 0.24218750 0.18359375 0.09179688 0.00000000 0.48242188
14 4 0.00000000 0.42578125 0.09179688 0.00000000 0.48242188
15 3 0.21289063 0.00000000 0.30468750 0.00000000 0.48242188
16 2 0.21289063 0.15234375 0.00000000 0.15234375 0.48242188
17 1 0.21289063 0.15234375 0.07617188 0.00000000 0.55859375
18 4 0.00000000 0.36523438 0.07617188 0.00000000 0.55859375
19 3 0.18261719 0.00000000 0.25878906 0.00000000 0.55859375
20 2 0.18261719 0.12939453 0.00000000 0.12939453 0.55859375
21 1 0.18261719 0.12939453 0.06469727 0.00000000 0.62329102
22 4 0.00000000 0.31201172 0.06469727 0.00000000 0.62329102
23 3 0.15600586 0.00000000 0.22070313 0.00000000 0.62329102
24 2 0.15600586 0.11035156 0.00000000 0.11035156 0.62329102
25 1 0.15600586 0.11035156 0.05517578 0.00000000 0.67846680
26 4 0.00000000 0.26635742 0.05517578 0.00000000 0.67846680
27 3 0.13317871 0.00000000 0.18835449 0.00000000 0.67846680
28 2 0.13317871 0.09417725 0.00000000 0.09417725 0.67846680
29 1 0.13317871 0.09417725 0.04708862 0.00000000 0.72555542
30 4 0.00000000 0.22735596 0.04708862 0.00000000 0.72555542
31 3 0.11367798 0.00000000 0.16076660 0.00000000 0.72555542
32 2 0.11367798 0.08038330 0.00000000 0.08038330 0.72555542
33 1 0.11367798 0.08038330 0.04019165 0.00000000 0.76574707
34 4 0.00000000 0.19406128 0.04019165 0.00000000 0.76574707
35 3 0.09703064 0.00000000 0.13722229 0.00000000 0.76574707
36 2 0.09703064 0.06861115 0.00000000 0.06861115 0.76574707
37 1 0.09703064 0.06861115 0.03430557 0.00000000 0.80005264
38 4 0.00000000 0.16564178 0.03430557 0.00000000 0.80005264
39 3 0.08282089 0.00000000 0.11712646 0.00000000 0.80005264
40 2 0.08282089 0.05856323 0.00000000 0.05856323 0.80005264
41 1 0.08282089 0.05856323 0.02928162 0.00000000 0.82933426
42 4 0.00000000 0.14138412 0.02928162 0.00000000 0.82933426
43 3 0.07069206 0.00000000 0.09997368 0.00000000 0.82933426
44 2 0.07069206 0.04998684 0.00000000 0.04998684 0.82933426
45 1 0.07069206 0.04998684 0.02499342 0.00000000 0.85432768
46 4 0.00000000 0.12067890 0.02499342 0.00000000 0.85432768
47 3 0.06033945 0.00000000 0.08533287 0.00000000 0.85432768
48 2 0.06033945 0.04266644 0.00000000 0.04266644 0.85432768
49 1 0.06033945 0.04266644 0.02133322 0.00000000 0.87566090
50 4 0.00000000 0.10300589 0.02133322 0.00000000 0.87566090
51 3 0.05150294 0.00000000 0.07283616 0.00000000 0.87566090
52 2 0.05150294 0.03641808 0.00000000 0.03641808 0.87566090
53 1 0.05150294 0.03641808 0.01820904 0.00000000 0.89386994
54 4 0.00000000 0.08792102 0.01820904 0.00000000 0.89386994
55 3 0.04396051 0.00000000 0.06216955 0.00000000 0.89386994
56 2 0.04396051 0.03108478 0.00000000 0.03108478 0.89386994
57 1 0.04396051 0.03108478 0.01554239 0.00000000 0.90941232
58 4 0.00000000 0.07504529 0.01554239 0.00000000 0.90941232
59 3 0.03752264 0.00000000 0.05306503 0.00000000 0.90941232
60 2 0.03752264 0.02653252 0.00000000 0.02653252 0.90941232
61 1 0.03752264 0.02653252 0.01326626 0.00000000 0.92267858
62 4 0.00000000 0.06405516 0.01326626 0.00000000 0.92267858
63 3 0.03202758 0.00000000 0.04529384 0.00000000 0.92267858
64 2 0.03202758 0.02264692 0.00000000 0.02264692 0.92267858
65 1 0.03202758 0.02264692 0.01132346 0.00000000 0.93400204
66 4 0.00000000 0.05467450 0.01132346 0.00000000 0.93400204
67 3 0.02733725 0.00000000 0.03866071 0.00000000 0.93400204
68 2 0.02733725 0.01933035 0.00000000 0.01933035 0.93400204
69 1 0.02733725 0.01933035 0.00966518 0.00000000 0.94366722
70 4 0.00000000 0.04666760 0.00966518 0.00000000 0.94366722
71 3 0.02333380 0.00000000 0.03299898 0.00000000 0.94366722
72 2 0.02333380 0.01649949 0.00000000 0.01649949 0.94366722
73 1 0.02333380 0.01649949 0.00824974 0.00000000 0.95191696
74 4 0.00000000 0.03983329 0.00824974 0.00000000 0.95191696
75 3 0.01991665 0.00000000 0.02816639 0.00000000 0.95191696
76 2 0.01991665 0.01408320 0.00000000 0.01408320 0.95191696
77 1 0.01991665 0.01408320 0.00704160 0.00000000 0.95895856
78 4 0.00000000 0.03399984 0.00704160 0.00000000 0.95895856
79 3 0.01699992 0.00000000 0.02404152 0.00000000 0.95895856
80 2 0.01699992 0.01202076 0.00000000 0.01202076 0.95895856
81 1 0.01699992 0.01202076 0.00601038 0.00000000 0.96496894
82 4 0.00000000 0.02902068 0.00601038 0.00000000 0.96496894
83 3 0.01451034 0.00000000 0.02052072 0.00000000 0.96496894
84 2 0.01451034 0.01026036 0.00000000 0.01026036 0.96496894
85 1 0.01451034 0.01026036 0.00513018 0.00000000 0.97009912
86 4 0.00000000 0.02477070 0.00513018 0.00000000 0.97009912
87 3 0.01238535 0.00000000 0.01751553 0.00000000 0.97009912
88 2 0.01238535 0.00875776 0.00000000 0.00875776 0.97009912
89 1 0.01238535 0.00875776 0.00437888 0.00000000 0.97447800
90 4 0.00000000 0.02114311 0.00437888 0.00000000 0.97447800
91 3 0.01057156 0.00000000 0.01495044 0.00000000 0.97447800
92 2 0.01057156 0.00747522 0.00000000 0.00747522 0.97447800
93 1 0.01057156 0.00747522 0.00373761 0.00000000 0.97821561
94 4 0.00000000 0.01804678 0.00373761 0.00000000 0.97821561
95 3 0.00902339 0.00000000 0.01276100 0.00000000 0.97821561
96 2 0.00902339 0.00638050 0.00000000 0.00638050 0.97821561
97 1 0.00902339 0.00638050 0.00319025 0.00000000 0.98140586
98 4 0.00000000 0.01540389 0.00319025 0.00000000 0.98140586
99 3 0.00770194 0.00000000 0.01089219 0.00000000 0.98140586
100 2 0.00770194 0.00544610 0.00000000 0.00544610 0.98140586
101 1 0.00770194 0.00544610 0.00272305 0.00000000 0.98412891
102 4 0.00000000 0.01314804 0.00272305 0.00000000 0.98412891
103 3 0.00657402 0.00000000 0.00929707 0.00000000 0.98412891
104 2 0.00657402 0.00464853 0.00000000 0.00464853 0.98412891
105 1 0.00657402 0.00464853 0.00232427 0.00000000 0.98645318
106 4 0.00000000 0.01122255 0.00232427 0.00000000 0.98645318
107 3 0.00561128 0.00000000 0.00793554 0.00000000 0.98645318
108 2 0.00561128 0.00396777 0.00000000 0.00396777 0.98645318
109 1 0.00561128 0.00396777 0.00198389 0.00000000 0.98843706
110 4 0.00000000 0.00957905 0.00198389 0.00000000 0.98843706
111 3 0.00478952 0.00000000 0.00677341 0.00000000 0.98843706
112 2 0.00478952 0.00338671 0.00000000 0.00338671 0.98843706
113 1 0.00478952 0.00338671 0.00169335 0.00000000 0.99013042

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This is fewer squeezes than starting at zone 4 initially, which takes 116 squeezes.
  Posted by Charlie on 2004-02-17 08:41:23
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