2048 Quiz Answer

This puzzle is based on the very popular game 2048, which was invented by Gabriele Cirulli, and (if you haven't seen it) details can be found here.

As you play the game you gradually end up with tiles of higher and higher values - 2, 4, 8, 16, and so on, up in powers of two to the 2048 tile which gives the game its name. And beyond to higher powers of two if you are good (and lucky) enough.

Every time two tiles combine you score the value of the resulting tile. So combining two 2 tiles to make a 4 tile will score 4.

To make an 8 tile you will have to make two 4 tiles (scoring 4 each) and combine them to make the 8 tile (scoring 8) which makes a total of 16. So combining tiles to make an 8 tile will score 16.

Can you find a general formula to calculate the score for any power of two? That is, to make a tile of 2 to the power x will produce a score of y.

Each tile with a new power of two requires two tiles of the previous power of two, so each new power is twice the score of the previous power (to make the two tiles) plus the value of combining these (the new power).

As we've seen an 8 tile scores twice the 4-tile score plus 8, ie 16. A 16 tile scores twice the 8-tile score (2 x 16) plus 16, ie 48. A 32 tile scores twice the 16-tile score (2 x 48) plus 32, ie 128. And so on . . .

The scores for tiles with each increasing power of two are:

Tile 4 scores 4

Tile 8 scores 16

Tile 16 scores 48

Tile 32 scores 128

Tile 64 scores 320

Tile 128 scores 768

Tile 256 scores 1792

Tile 512 scores 4096

Tile 1024 scores 9216

Tile 2048 scores 20480

The general formula when n is the power of two is

$$ 2^n * (n - 1) $$

Devotees of the game will have spotted that we are ignoring the fact that sometimes the new tiles are 4s and not 2s, and so the scoring would be a little different. However, for the purposes of this puzzle, we are keeping it simple and just working in 2s.

If you find the puzzle easy you can always extend it, assuming that one new tile in seven is a 4 not a 2!

If one new tile in seven is a 4 then you will get six 2 tiles and one 4 tile, which adds up to 16.

In the solution above you combine eight 2 tiles to make four 4 tiles (scoring 16), and then you combine the four 4 tiles to make two 8 tiles (scoring another 16), and finally you combine the two 8 tiles to make a 16 tile (scoring 16 again). This results in the 48 that we have in the solution above.

With six 2 tiles and one 4 tile you combine the six 2 tiles to make three 4 tiles (scoring 12), and then using these three 4 tiles plus the other 4 tile you proceed as above, scoring 16 and 16. This makes a total of 44 instead of 48.

From 16 up through the powers of two you can proceed as before, namely twice the score for the previous power plus the power, remembering that the 16 tile scores 44 not 48. This starts

Tile 16 scores 44

Tile 32 scores 120

Tile 64 scores 304

And the general formula is

2^n * (n - 1) - 2^(n - 2)


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