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.)

...just kidding. There was nothing about this at the Pepperdine University website either. But when we are actually using a tube of toothpaste, we instinctively press from the end of the tube to the front opening. Is this the most efficient strategy? With three zones, it takes 48 squeezes to get 99.1% of the toothpaste onto the brush:

tube(1/3 1/3 1/3) brush(0)
press 1:tube(0 2/3 1/3)
press 2:tube(1/3 0 2/3)
3:tube(1/3 1/3 0) brush(1/3)
1:tube(0 2/3 0)
2:tube(1/3 0 1/3)
3:tube(1/3 1/6 0) brush(1/2)
1:tube(0 1/2 0)
2:tube(1/4 0 1/4)
3:tube(1/4 1/8 0) brush(5/8)
1:tube(0 3/8 0)
2:tube(3/16 0 3/16)
3:tube(3/16 3/32 0) brush(23/32)
1:tube(0 9/32 0)
2:tube(9/64 0 9/64)
3:tube(9/64 9/128 0) brush(101/128)
1:tube(0 27/128 0)
2:tube(27/256 0 27/256)
3:tube(27/256 27/512 0) brush(431/512)
1:tube(0 81/512 0)
2:tube(81/1024 0 81/1024)
3:tube(81/1024 81/2048 0) brush(1805/2048)
1:tube(0 243/2048 0)
2:tube(243/4096 0 243/4096)
3:tube(243/4096 243/8192 0) brush(7463/8192)
1:tube(0 729/8192 0)
2:tube(729/16384 0 729/16384)
3:tube(729/16384 729/32768 0) brush(30581/32768)
1:tube(0 2187/32768 0)
2:tube(2187/65536 0 2187/65536)
3:tube(2187/65536 2187/131072 0)
brush(124511/131072)
1:tube(0 6561/131072 0)
2:tube(6561/262144 0 6561/262144)
3:tube(6561/262144 6561/524288 0)
brush(504605/524288)
1:tube(0 19683/524288 0)
2:tube(19683/1048576 0 19683/1048576)
3:tube(19683/1048576 19683/2097152 0)
brush(2038103/2097152)
1:tube(0 59049/2097152 0)
2:tube(59049/4194304 0 59049/4194304)
3:tube(59049/4194304 59049/8388608 0)
brush:(8211461/8388608)
1:tube(0 177147/8388608 0)
2:tube(177147/16777216 0 177147/16777216)
3:tube(177147/16777216 177147/33554432 0)
brush(33022991/33554432)
1:tube(0 531441/33554432 0)
2:tube(531441/67108864 0 531441/67108864)
3:tube(531441/67108864 531441/134217728 0)
brush(132623405/134217728)
1:tube(0 1594323/134217728 0)
2:tube(1594323/268435456 0 1594323/268435456)
3:tube(1594323/268435456 1594323/536870912 0)

brush(532087943/536870912)
= 99.1%

Posted by Penny on 2004-02-17 07:27:09