Balancing Puzzle - help needed

[indiekid]

Hi everyone

I'm asking the community's help in solving a puzzle which will enable me to balance a tabletop game I'm designing. If you like sudoku-type puzzles, this might be for you. I do not think there is an exact solution to the puzzle, and it's easy to create a workable one by hand. My curiosity won't be satisfied, however, until I have found the optimal solution to it. This may require computation. Either way it is beyond me.


Click here to view the spreadsheet. Please don't edit it – it should be set to allow you to download. To share your answer – and you could have multiple goes – please upload as a comment here the content of the "Card" column and the "Variables" column. The latter will populate automatically as you complete the puzzle, if you use the spreadsheet as it is.

The game consists of 22 cards. Each has a numerical value. Each also shows between 1 and 3 variables from a, b, c and d. For example, the card of numerical value 12 already contains symbols b and d. The greater the card's numerical value, the more influence the variables printed will have on the state of play. I have denoted the total influence each variable has, ie. the sum of each of the cards it appears on, with the symbols Σa, Σb, Σc, and Σd.

The challenge:
Complete the grid by inserting the letter a, b, c or d into each of the blue squares, such that:
1.   No variable appears twice on the same card.
2.   All variables appear an equal number of times (twelve).
3.   Three of the values of Σa, Σb, Σc, and Σd are the same.
4.   The fourth Σ value from the above is close to the others but very slightly higher
5.   That variable does not appear on the card with a numerical value of zero

Σa, Σb, Σc, and Σd should update for you every time you add a letter into the grid. Needless to say, the variables already populated on the grid and currently in white squares cannot be changed. In (3) above I don't believe we can get all three values equal. Also the definition of "close" in (4) is vague.

Thanks in advance for your help here. I'm just as interested in your methodology. Is there, for example, a branch of mathematics which will provide us with an algorithm for finding the optimum solution? Can an iterative process do it? Or can we just solve it by visual inspection as we work from top to bottom, or bottom to top?

When we've had a few goes I'll tell you a bit more about the project. As a variant, is it possible to fill the grid such that Σa, Σb, Σc, and Σd are all equal? There is a second, very similar, puzzle to solve afterwards too.

Cheers for reading and I hope you'll find this interesting.

indiekid