The six passes
A pass does not take an amount of soil off a surface. It takes a share of what is still there. So the second pass is worth less than the first, the fifth is worth almost nothing, and the honest question about any job is not how hard we scrub but how many passes the surface needs.
r(n) = (1 − k)ⁿ
r is the share of the original soil still on the surface, n is the number of passes, and k is the share one pass removes. k is a property of the surface, not of the crew or the product. Our published finish line is r ≤ 0.05.
The colours on this page are the same law
There are six colours on this site and not a seventh, and none of them was chosen. Each one is one surface at a residual on the curve above, at a reference rate of k = 0.50, mixed in linear light. The ink you are reading is that surface at 100 per cent residual. The page behind it is the same surface at zero. The four in between are the passes, and the reason the last three are almost impossible to tell apart from the ground is not a printing fault — it is what a halving law looks like.
- pass 0r 1.0000#12160f
- pass 1r 0.5000#7cb2ae
- pass 2r 0.2500#95d5d1
- pass 3r 0.1250#9fe4df
- pass 4r 0.0625#a4ebe6
- pass ∞r 0.0000#a9f2ed
| Surface | k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | finished |
|---|---|---|---|---|---|---|---|---|---|
| Glazing | 0.82 | 1.000 | 0.180 | 0.032 | 0.006 | 0.001 | 1.9e-4 | 3.4e-5 | 2 |
| Painted wall | 0.61 | 1.000 | 0.390 | 0.152 | 0.059 | 0.023 | 0.009 | 0.004 | 4 |
| Hard floor | 0.55 | 1.000 | 0.450 | 0.202 | 0.091 | 0.041 | 0.018 | 0.008 | 4 |
| Tile grout | 0.31 | 1.000 | 0.690 | 0.476 | 0.329 | 0.227 | 0.156 | 0.108 | 9 |
| Sealed limestone | 0.00 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | never |
The one we refuse
Sealed limestone has a k of zero on our card, and the zero is not a measurement of our cloth. Wet detergent cleaning strips the seal, which is the part doing the protecting, so the correct number of passes is none. We turn the work down and tell the building to dry-dust it. The pair below is the same slab, printed the same way, with only the treatment's second step separating them.
What this changes on the invoice
Where k is high the number of passes is known before we start, so the work is priced per unit: glazing by the pane, hard floor by the square metre. Where k is low it is not knowable in advance, so it is timed. Grout at k = 0.31 is still at 0.108 after six passes and needs 9 to reach the finish line. Charging that by the square metre would mean either overcharging the easy rooms or losing money on the hard ones.