Deli problem

[David Wilson]

I work at a deli now, but I'm always thinking math, so here goes.

The scales at my deli rounds down weights to the nearest one hundredth of a pound (yeah, it's an odd mix of U.S. and metric, but I guess it's standard for U.S. delis). So 0.778 pounds of meat or cheese would weigh out as 77 hundredths of a pound.

Suppose I am slicing a block of cheese that happens to yield perfectly uniform slices weighing precisely 0.03871 pounds each. I start with an empty scale, and add slices one at a time, staying less than a pound. The successive readings on the scale are:

0 3 7 11 15 19 23 27 30 34 38 42 46 50 54 58 61 65 69 73 77 81 85 89 92 96

The next reading would be 100, which is a pound or over, so I stop there. Similarly, if the slices weighed 0.04137 pounds each, I would get the readings

0 4 8 12 16 20 24 28 33 37 41 45 49 53 57 62 66 70 74 78 82 86 91 95 99

of if the slices were a very heavy 0.29337 pounds, I would get the readings

0 29 58 88

My questions are:

Given slices of perfectly uniform positive weight, how many different reading sequences are possible? How are they distributed by number of readings?






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