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 Total Toothpaste (Posted on 2004-02-16)
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.)
 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

```

----
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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