What can be asked

Why Question Counts Collide Less

Four hundred and five distinct question counts against one hundred and ninety-five cell counts, from the same schemas. The difference is a prime.

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The same four hundred and sixty-one schemas produce one hundred and ninety-five distinct cell counts and four hundred and five distinct question counts.

More than twice as many distinct values from the second figure, computed from exactly the same inputs. The reason is arithmetical and worth setting out, because it says which figure to trust when only one is available.

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The factors each product can contain

Cell counts are products of numbers drawn from two, three, four, five and six.

Question counts are products of numbers drawn from three, four, five, six and seven.

The ranges overlap in three places and differ at both ends. What matters is what those numbers break down into.

Facet size	Contributes to cells	Contributes to questions
2	2	3
3	3	4 = 2·2
4	4 = 2·2	5
5	5	6 = 2·3
6	6 = 2·3	7

The right-hand column contains a seven. The left-hand column contains nothing larger than five, and five appears only once.

What the seven does

A seven cannot be built from twos, threes and fives. So a question count containing a seven can only have come from a schema containing a facet of six values, and the number of sevens in it is the number of such facets.

That single fact partitions the record. Every schema announces, in its question count, exactly how many six-value facets it carries, and no rearrangement can disguise it.

Cell counts have no equivalent marker. Six is two times three, so a facet of six can be swapped for a facet of two plus a facet of three with the product unchanged — which is precisely the manoeuvre that produces the large collision classes.

The collisions in the cell counts come from six being composite. The absence of collisions in the pair comes from seven being prime.

The sizes of the largest classes

The largest group of schemas sharing a cell count holds seven. Four different cell counts are reached that often: one hundred and forty-four, two hundred and eighty-eight, four hundred and thirty-two, and five hundred and seventy-six.

The largest group sharing a question count holds three. One question count is reached that often: one thousand two hundred and ninety-six, by four filters of five, by five filters of two, two, three, five and five, and by six filters of two, two, two, two, three and three.

Seven against three. The second figure is more than twice as sharp at its worst point.

Which figure to use when only one is possible

The question count, if the choice is free.

It distinguishes more than twice as many structures, its worst ambiguity is a group of three rather than seven, and it carries the count of six-value facets in plain sight.

There is one caution and the page on one-choice filters makes it: a filter with a single option doubles the question count while adding nothing. So a question count quoted alone can be inflated by inert apparatus in a way a cell count cannot.

In practice that cuts both ways and does not change the recommendation. A cell count quoted alone can be inflated by a facet whose values nobody uses, which is both commoner and harder to spot.

The two primes, side by side

Three and seven are the only primes a question count can contain that a cell count cannot.

Seven comes from a facet of six. Three comes from a facet of two — but threes also arrive inside the cell counts, via facets of three and six, so the three marks nothing.

The seven marks everything. Count the sevens in a question count and you have counted the six-value facets, exactly, with no ambiguity and no alternative explanation.

That is the single arithmetical fact behind the whole difference in discriminating power between the two figures, and it is why the question count is the sharper of the two.

The honest conclusion

Neither figure alone is reliable and one of them is more reliable than the other.

If a board is to be described by a single number, the question count is the better single number, by a factor of two in discriminating power. If it is to be described properly, both figures go in, the description becomes exact, and the argument about which one to pick stops mattering.

The whole of this section has been circling the same point from different angles, and it is the same point the record keeps returning to: the second figure is almost free, and nearly nobody takes it.