# 1,000,000 cubes optimally solved in both QTM and FTM

I have solved all 1,000,000 random cube positions from the earlier article now in both QTM and HTM. Here is the resulting grid:
```    12h 13h 14h  15h   16h    17h    18h   19h     sum
15q   1   1   3    2     -      -      -     -       7
16q   -   2  18   48    35      -      -     -     103
17q   -   3  23  143   347    354      -     -     870
18q   -   5  40  305  1713   4520   2034     -    8617
19q   -   1  40  505  5190  29711  33363   474   69284
20q   -   2  39  674  9932 100164 212466  7213  330490
21q   -   -   9  345  7697 104052 301668 16371  430142
22q   -   -   -   41  1533  28173 120449  9720  159916
23q   -   -   -    -     1     53    427    90     571
sum   1  14 172 2063 26448 267027 670407 33868 1000000
```
This is a partial answer to a previously posted question; it appears about 2034 cubes out of a million are 18q* and 18f* simultaneously, and about 2899 out of a million cubes have the same distance in QTM and HTM; this is about one in every 345 positions.

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### what is QTM and FTM ?

what is QTM and FTM ?

### QTM and FTM

QTM = Quarter Turn Metric
FTM = Face Turn Metric

In mathematics, a metric is a "distance function" that can literally be geometric distance, but that can also be a more abstract function that can give the concept of distance to many sets, including sometimes to non-numerical sets. If d is a metric, the key attribute is the triangle inequality: d(x,z) ≤ d(x,y) + d(y,z) - the distance from x to z must be less than or equal the distance from x to y plus the distance from y to z. The triangle inequality is the attribute that most contributes to a metric being analogous to a geometric distance. A metric must also satisfy d(x,y) = d(y,x) and d(x,y) = 0 if and only if x = y.

In Rubik's cube, the distance being measured is the number of moves between two positions. For example, we have d(x,y) = 12 if the minimum number of moves to get from position x to position y is 12 moves. Most often (but not always), the distances that are being measured are the distances between the Start position and some particular position x. Usually, d(Start,x) is called the length of x and sometimes is written as |x|.

Quarter turn metric refers to counting the twelve quarter turns as moves. Face turn metric refers to counting the twelve quarter turns plus the six half turns as moves.

In the quarter turn metric, F is one move and we might say |F|=1, and FF is two moves and we might say |FF|=2. In the face turn metric, F is still one move and we might say |F|=1, and FF is one move usually written as F2 or as F2 and we might say |F2| = 1.