Friday, March 14, 2014

The Christmas Investigations - Turnabout: Part Two

Christmas Investigations
Part Four
In the first of our Christmas Investigations we examined the strategic word game, Turnabout and tried to get to grips with its complex scoring method. We tried out The Sphere Game and The Star Game and in this post we'll aim to answer the first Christmas Investigation:

What is the maximum score on a Turnabout grid, and how can it achieved?

You might remember that we set you the task of filling a grid of grey spheres completely with ones in your choice of player colour. Each time you made a new row of three you scored five points, each time you made a new row of four; ten points was your reward.

I asked you what the theoretical minimum and maximum scores were, given you had to turn every sphere to your colour. I also asked you whether these scores were practically achievable.

The first thing we should work out is the theoretical minimum and maximum scores. To get this we should work out the total of every scoring line on the board. There is a complication however, making a scoring line in some places prevents you from making others.

First things first, some scoring is unavoidable. If you fill a line of four, anywhere, you score ten points. Since you have to fill the whole grid, you have to score the ten point lines for every row, column and diagonal.

There are actually four other unavoidable scores in this four by four grid, and these are the four shorter diagonals. 

These 14 scores, a total of 120 points, comprise all the unavoidable ways to score when filling the whole board. Thus the minimum theoretical score is 120 points.

There are other ways to score too, you may score a five point line from either the first three spheres in a row, or the last three, but not both as the fourth sphere placed in a row will score a ten point line instead.

You may have realised from this that there are two different scores you can get by filling four spheres in any row, column or diagonal.

If you fill in the first three, then the fourth or the last three and then the first; you will score first a five point line, then a ten point line for a total of fifteen. If you fill in the first and last to begin with however, you can only score the ten point line; as this supersedes the five point one. 

Of these twenty lines only ten are possible to score in any one game, the fourth sphere needed to score each of the other ten would instead make a line of four and score ten. Thus there are ten multiplied by five avoidable points in this grid, making a total of fifty points. Thus the theoretical maximum score for this grid is 170 points.

So it stands to reason that if we want to score the minimum possible score of 120, we have to avoid scoring on an of the twenty avoidable lines. To do this we need to place both ends of each row, column and diagonal in the grid, before we place both of the centre spheres.

The best way I have found to do this is to place the spheres in two spirals. First placing the four corners.

Then placing one adjacent sphere to each corner, before placing two diagonally opposite spheres in the centre block of four squares.

The last of these ten spheres is the first to score. At this stage almost all the avoidable scores have been blocked. The next two moves completing the first and last columns remove the possibility of scoring from the remaining avoidable lines.

Filling in the last few spheres causes the scores to rack up considerable, but these are all unavoidable scores, even the final move which scores 35. Indeed all routes I have found to the minimum score have a final placement worth 35 points.

Oddly enough, or maybe not, the solution I found for achieving the maximum score also works on spirals. This time the trick is starting from the middle and working to the outer edge, making all the avoidable scoring lines as you go.

Just like before our first four spheres do not score, but this is where the similarity ends. Here we need to score quickly, and we need to score on the avoidable threes before we score on the related fours.

To do this we work symmetrically placing one sphere on each side of our opening square. In doing this we score four of the avoidable five point lines.

The next step is to place another sphere on each side of the square, this scores on the four point lines that relate to the avoidable threes scored earlier, and scores the four unavoidable threes.

Finally we place spheres in each of the four corners, starting with two that are diagonally opposite. each move scores more points that the last, and although we don't have such a big score for the final move, we have been scoring for longer, and with more consistency. We have scored the maximum of 170.

So we have discovered that the secret strategy to a word game about spheres is to work in spirals. As it happens the way to score the maximum in a three by three grid is also to place the spheres in a spiral. Suppose we had a new rule that gave a 20 point score for a line of five? The answer for the maximum is still a spiral, but the minimum gives rise to an interesting pattern. 

Have a play with spheres and see what you can discover, and if Turnabout ever makes it back to our screens - you have a perfect strategy.

That's all for today, be sure to follow me on Twitter, it's @WanderingPuzz, and until next time; keep puzzling!

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