Three ends

Reading The Half-Grid

The three-end table is a triangle, not a square, and the diagonal means something specific. How to find any arrangement in one step.

The three-end table is a triangle. Rows are the gap to the second end, columns the gap to the third, and only the cells where the column is at or past the row are filled in.

The missing half is not missing data. It is the same data, written the other way round.

If screenshots or application records are considered, workload reporting for global teams should be tested with clear notice, limited retention and settings proportionate to the scheduling problem. A useful external reference for the surrounding topic is SHRM.

A narrower window, read closely

every cell prints the hours all three share
reversed: no hour at all

28 arrangements drawn; no of them leave no hour the three can all attend; the grid is a half: the two gaps are interchangeable, so the other half would repeat it.

The leading diagonal holds the strongest arrangement at each distance; the value falls by one for every hour the two outer ends move apart.

Computed from the two gaps. The working day is a stated convention; the intersection is arithmetic.

Why it is a half

An arrangement in which the second end is at a gap of two and the third at a gap of five is identical to one in which the second is at five and the third at two. The three ends are in the same three places; the only difference is the order in which two of them were listed.

Filling in both halves would have produced a symmetric square in which every arrangement appeared twice. That is twice the ink for no additional fact, and it invites the reader to look for a difference between the halves that does not exist.

So the grid is drawn as a triangle, and the legend under every such drawing says why, because an empty half with no explanation reads as a hole in the record rather than as a deliberate economy.

What the diagonal means

The leading diagonal — where both gaps are equal — is the case where the second and third ends are in the same hours as each other. Effectively there are two distinct positions and one of them has two people in it.

Those are the strongest arrangements at any given distance, and reading down the diagonal gives the two-end answer directly: eight hours at a gap of zero, seven at one, six at two, and so on.

The diagonal is the two-end problem hiding inside the three-end one. Everything off the diagonal is the two-end problem with the ends pulled further apart.

Finding an arrangement in one step

Pick any one of the three ends as the reference and express the other two as gaps from it.
If the smaller gap is the row and the larger the column, the cell is already where you expect. If not, swap them — the answer is the same.
Read the printed value.

The choice of reference end does not matter. Any of the three produces a pair of gaps that lands on the same answer, because the answer depends only on the spread of the three and not on which one is called first.

Why the value is printed in the cell

So that the grid can be read without a key, in greyscale, and by anybody who does not have colour available to them.

A grid of this kind is ordinarily shaded, with a scale printed alongside, and that construction fails completely when the colour does not arrive. Printing the figure costs a few pixels of width per cell and removes the dependency entirely. The one distinction the grid does carry visually — an arrangement that shares nothing — is marked by reversing the cell, which survives any reproduction that preserves light and dark.

What a grid does that a list cannot

The same ninety-two surviving arrangements could have been printed as a list of triples with their values. The grid is better, and for a reason worth naming because it applies to every table of this kind.

A list answers lookups. A grid answers lookups and shows the shape of the answer: where the values are high, where they fall away, where the boundary runs. The band of survivors around the diagonal is visible in the grid at a glance and would be invisible in a list of the same numbers.

That shape is the finding. The individual values are reference material.

Why reversal rather than shading

The one visual distinction the grid carries is between an arrangement that shares nothing and one that shares something. It is drawn by reversing the cell — dark becomes light, light becomes dark — rather than by tinting it.

Three reasons, in order of how often they bite.

It survives greyscale. A tinted cell in a scale of greens is indistinguishable from its neighbour once the colour is gone, and colour goes more often than people expect: printed, photocopied, projected badly, viewed by somebody who does not see the distinction.

It survives being small. A reversed cell is legible at any size at which the grid is legible at all. A tint two steps along a scale is not.

It cannot be misread as a value. A tint implies a quantity, and a reader will try to estimate it. A reversal implies a category, which is what is being shown: these cells are not a small amount, they are none.

Reversal says "this one is different". Shading says "this one is somewhat less". When the distinction is categorical, saying it in a language of degree is a mistake however pretty the result.

The limit of the grid

It holds whole-hour gaps only, and it holds the standard shape at all three ends. Three ends with different shapes, or gaps that are not whole hours, are both outside it. The first is not in this record at all; the second is the subject of the next section.