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Son Of Darts

The Darts Problem

Suppose that you have a dartboard that is divided into R regions. Each dartboard region has a positive integer value associated with it. Further suppose that you have D darts and that you throw each of them at the dartboard. Each dart either lands in one of the board's R regions or misses the board altogether. Your score is the sum of the values for the regions in which the darts land. A dart that misses the board contributes nothing to your score. If multiple darts land in the same region, you accumulate the value for that region multiple times.

For example, suppose that R = 5, that the dartboard regions have values (1, 2, 4, 7, 11), and that D = 3. If your three darts land in regions 2, 4 and 11 you score 17 points. If one dart misses the board and the other two land in region 7 you score 14 points.

The Darts Problem is this: for a given R and D, determine what values should be associated with a dartboard's R regions in order to maximize the smallest score unattainable by throwing D darts.

For example, again suppose that R = 5 and D = 3. If you choose the values (1, 2, 4, 7, 11) for the board's 5 regions, then the smallest score unattainable with your 3 darts is 27. But if you choose the values (1, 4, 6, 14, 15) the smallest unattainable score is 37. The second choice of values is therefore a better choice for 3 darts and 5 regions.

By the way, the Darts Problem is more commonly known as the Postage Stamp problem, wherein you are asked to determine which R denominations of stamps to issue in order to maximize the smallest postage unattainable with D stamps.

The Contest

Submit (see How to Enter, below) your best solutions to the Darts Problem for the following values of R and D:

D = number of darts R = number of dartboard regions
3 1 through 40
4 1 through 30
5 1 through 20
6 1 through 10
Thus, there are 100 problems and you are asked to submit 100 solutions. You can submit more than one solution for the same problem, but if you do we count only your best solution. There is no penalty for submitting multiple solutions for the same problem.

See The Scoring System, below, to learn how we determine the winner.

The Prizes

The prizes are metal sculptures by Bathsheba Grossman. The 1st place winner has his pick from among the larger ones, and 2nd place chooses from the smaller.

How to Enter

Just paste your solutions into the large box on the Submit page and click Submit. Format your solutions as follows:

  • An individual solution consists of
    • a number (representing the number of darts), followed by
    • a colon, followed by
    • a comma-delimited list of region values.
  • Submit multiple solutions in a single batch by separating them with semicolons. Do not place a semicolon after your last solution.
  • Include spaces and line breaks anywhere you like (except within a number) to improve readability.

There's an example of an entry in the Frequently Asked Questions section, below.

Do not submit entries under more than one account. This is important. Do not submit entries under more than one account.

The Scoring System

Each time you submit a solution we will merge it with your prior solutions, if any. The result will be a virtual entry containing your best solutions for each of the 100 problems. We will give each of these 100 solutions a subscore from 0 to 1 and their sum will be your contest score.

We score the individual solutions as follows. If your solution is the best that was submitted for that problem, we give it 1 point; otherwise we give it only a fraction of a point. The fraction is the solution's smallest-unattainable-score divided by the best smallest-unattainable-score submitted by anyone for that same problem.

In case of a tie, preference is given to the entrant whose latest improved solution was submitted least recently.

Let's walk through a simplified example. Suppose that we reduce the contest to only 3 problems -- finding the dartboard regions that maximize the smallest unattainable score in these cases:

4 darts, 2 regions
4 darts, 3 regions
4 darts, 4 regions

Further suppose that we have only 3 entrants (Dana, Pat and Syd) and that these are their best solutions for each of the 3 problems:

Dana Pat Syd
4 darts, 2 regions 1, 2 1, 3 1, 4
4 darts, 3 regions 1, 4, 7 1, 2, 3 1, 3, 5
4 darts, 4 regions 1, 3, 5, 7 1, 4, 7, 10 1, 2, 3, 4

The smallest unattainable scores corresponding to these solutions are:

Dana Pat Syd Best Solution
4 darts, 2 regions 9 11 11 11
4 darts, 3 regions 17 13 17 17
4 darts, 4 regions 23 23 17 23

The total scores for each entrant are computed as follows:

Dana Pat Syd
4 darts, 2 regions 9/11 = 0.82 11/11 = 1.00 11/11 = 1.00
4 darts, 3 regions 17/17 = 1.00 13/17 = 0.76 17/17 = 1.00
4 darts, 4 regions 23/23 = 1.00 23/23 = 1.00 17/23 = 0.74
Total 2.82 2.76 2.74

Finally, the standings page listing is as follows:

Rank Score Contestant
1 2.82 Dana
2 2.76 Pat
3 2.74 Syd

My Lawyer Would Want Me to Say This

I reserve the right to discontinue the contest at any time. I reserve the right to disqualify any entry or entrant for any reason that suits me. I reserve the right to interpret the rules as I see fit. I reserve the right to change the contest rules in mid-contest. In all matters contest-related, my word is final.

Frequently Asked Questions

Can you give an example of an entry?

Here's an entry that contains a single solution. It's for 3 darts and 7 regions. 3: 1,2,3,5,8,13,21

Here's another entry that contains solutions for 5 of the problems, some of which are for 4 darts and some of which are for 5. Note that the last solution is not ended with a semicolon. 4: 1; 4: 1,2; 4: 1,3,4; 5: 1,2; 5: 1,2,3

The immediately preceding solution could also be split across 5 lines for added readability. 4: 1;
4: 1,2;
4: 1,3,4;
5: 1,2;
5: 1,2,3

Can teams enter the contest?

Collaboration is allowed. However, only one of the collaborators may register. If two contestants are found to have collaborated, even if this occurred before one or both registered, both will be disqualified.

What information about my solutions can I share in the discussion group?

There are two types of information that you are forbidden to post. The first is specific solutions. The second is code. You may post scores, so if you want to tell everyone that you got a raw score of 4999 for (3 darts, 20 regions), go right ahead. You may also discuss the algorithms you are using.

How can I find out my subscores for the 100 individual problems?

You can't. Seriously.

Can I enter the contest more than once, using different names?

No. Nor can you enter more than once using the same name.

How do you know which questions are going to be asked frequently, before any questions have been asked?

It's a gift.