# The Bitcoin Field Puzzle

Time for another brain burner!

This puzzle is based on my game Tau (2011). TAU was evolved from another game of mine called OMEGA (2010) as a tool to help me teach my students how to multiply in binary base. It exploits a mechanic called ‘multiplicative scoring’.

Let’s start by drawing a square grid of any size (10x10, for example) and filling it with coins, one on each cell. This is our bitcoin field.

Now let’s split the field in two, by drawing a horizontal line like this:

Two groups of coins (above and below the line) have been created. In order to calculate our score, we multiply the number of coins of each group: S = 30 x 70 = 2.100 points.

Now let’s draw a vertical line like this:

Notice that now there are 4 groups. The new score is: S = 12 x 18 x 28 x 42 = 254.016.

By drawing this second line, we’ve managed to increase the score significantly. If we keep on drawing lines (creating groups) the score will keep on increasing for a while, but if we draw too many lines, it will start dropping!

In fact, if we draw all possible lines, the score will drop to 1 (100 groups of size 1 each)!

Notice also that drawing no lines produces a score of 100.

If a set of horizontal and/or vertical lines produces the highest possible score, this set is called a ‘solution’.

Challenge 1: Find a solution for a 10x10 field.

Challenge 2: Find a solution for a 1.000.000 x 1.000.000 field. Describe it instead of drawing it!

Let’s introduce an interesting twist now. What if some of the cells were empty?

Challenge 3: Find a solution for the field in figure 5. Empty groups have a value of zero!

And another twist: let’s mix bitcoins and dollars.

In this case, we define ‘solution’ as the partition that produces the highest possible bitcoin score and the lowest possible dollar score.

Challenge 4: Does a solution exist for the field in figure 6? If it does, can you find it?

Challenge 5: Find a partition for the field in figure 6 so that both the bitcoin and dollar scores are maximum.