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We need to ship 100 kits, each containing three beads of different colors. There are four colors: purple, blue, green, silver. We have the following quantities for each respective color: 53, 53, 85, 53. What are the fewest beads we will need to order, and what colors should we order to ship 100 kits with three beads in each kit and no kit having two or more beads of the same color? Also, how did you solve?

My guess was 56 total beads, using two colors: 41 of one, 15 of another, but I'm not sure if I solved it correctly:

  • make 53 boxes, using purple, blue, green, leaving you with 32 green and 53 silver.
  • order 15 green and 41 of either purple or blue, allowing you to make 100 kits using the three extra silver you have.

Now, that I think about it. I think this way has 6 boxes with duplicate silvers. :/

RubeOnRails
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Hint: how many total beads do you have? How many do you need to ship? Is there any reason you can't order just enough beads, like having too many of one color?

Added: You are correct that you need to order $56$ more beads. There is no best way. I would order $5$ green and $17$ of each other color, giving me $70,70,90,70$ just because I like round numbers. Now $30$ kits of each of $pbg, pgs, bgs$ and $40$ kits of $pbs$ get you there.

Added2: Any collection of $300$ beads can fill your order as long as you don't have more than $100$ of any one color. Add white beads to each bin to make a total of $100$ beads per bin. Now pack kits, taking one bead from each bin, with exactly one white bead in each kit. Remove the white beads and ship the kits.

Ross Millikan
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  • Total beads is the total of the four colors: 53 + 53 + 85 + 53 = 244 We need to ship 300 beads, one in each box, but no box can have any matching beads. Each box must have three beads of different colors. We can order as many beads as we want, but clearly we would like to order the least number possible. – RubeOnRails Oct 30 '13 at 00:22
  • Nice! What way would minimize the number of colors we need? – RubeOnRails Oct 30 '13 at 00:30
  • You can't buy all one color as that would give you more than $100$ of one color. But you can buy just two, as long as neither total exceeds $100$. See above for the proof, which I think is cute. – Ross Millikan Oct 30 '13 at 01:31