Remember the old-fashioned bars of soap? You are trying to figure out whether one would be enough to give to your son for four weeks of the sleep-away summer camp. I know, most of them never wash there, but let's hope that this time he will. You notice that when you leave this soap in the kids' bathroom and check in a week, all of its dimensions seem to erode by two: it has only half of the original height, half of the original width, and half of the original depth. But at home both of your kids are using it, in the camp it will be only for your son. How can you figure whether to pack one or two bars?

(this puzzle is adapted from a book by C. Koval)

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## Tuesday, April 20, 2010

### Packing for a sleep-away camp

Posted by
Maria
at
2:49 PM

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## 5 comments:

Assumptions:

(1) he will bathe at much at camp as at home

(2) each kid is using the same amount of soap

(3) they use the entire bar, and use an equal amount of soap each time they bathe

If the soap was originally 2x4x1 then it would have a volume of 8. The 1x2x1/2 bar left after a week has a volume of 1 (obviously, reducing all tehe dimensions in 1/2 has the effect of using it by 7/8). So in 7 days, they used 7/8 of the bar. In the next day, they will use the rest of the bar. So it takes 8 days for 2 kids to use 1 bar, or 16 days for 1 kid to use 1 bar.

If my son will be gone for 4 weeks (or 28 days), he's going to need 2 bars.

Sorry, that bit in parenthesis should have read:

(obviously, reducing all three dimensions in 1/2 has the effect of using the whole bar to 1/8 of its original size - or using up 7/8 of it)

Initial Volume V0 = l * w * h

after one week at home Volume =

(1/2)*l * (1/2)*w * (1/2)*h = (1/8)*V0

So two kids use 7/8 of a bar in one week.

Assuming that they use the same amount of soap in one week (in my house that is NOT the case as we found out when one child started using a different shower...) your son will use half of the 7/8 of a bar in one week, or 7/16 of a bar in one week.

(4 weeks)*(7/16 bar/week) = 28/16 bars of soap needed at camp => you need to pack at least 2 bars of soap (hey, he could accidentally drop one in the latrine!)

I'm thinking that the "volume of soap used" may be red herring, not quite relevant to this problem. Height alone, that might do it. The two children reduce the height by 50% in Week One. Assuming (!) the shower frequency is the same, it will take 2 weeks for one child to accomplish that wear. (The volume, incidentally, has now been reduced by 7/8 from the original. About 13% remains.)

And still, therefore, whether we consider just the height, or the volume, the camper will need an extra bar by the end of the camp. (They don't work so well covered in pine needles, either.) Tom

Kim got here first and solved it correctly - a puzzle point for her. Remember to pack two bars!

KJ and Tom - these are great and funny insights. Sounds like it is a good idea to have a separate bar soap for every kid at home, to see who is really washing and who is just pretending.

And those pine needles..... brrrr. It reminds me of a non-slip soap in a net that I bought once. There are should be some other creative camping solutions.

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