Change the layout, keep the function

Can the shape change while answers stay the same? · 0:00

Can the shape change while answers stay the same? Take the exclusive-or of two checks: A AND B, and C OR D. This four-input rule has sixteen assignments. We will move variables between the repository row and column groups, track an assignment through square and rectangular layouts, and check that its Boolean answer is preserved.

Count the rows and columns · 0:24

Each row variable doubles the number of row combinations. Two row variables give four rows. Two column variables give four columns. That is our familiar four-by-four layout, containing sixteen output entries altogether.

Move B to the column group · 0:38

Now put only A in the row group, and put B, C, D in the column group, in that order. One row bit gives two rows. Three column bits give eight columns. The new two-by-eight rectangle still has sixteen entries.

Follow the same assignment · 0:52

Follow one-zero-one-one again. A alone gives row index one. The remaining bits, B C D, are zero-one-one. Their weights are four, two, and one, so the column index is three. The selected entry remains one.

Move C to the row group instead · 1:07

For an eight-by-two layout, use A, B, C as row variables and D as the column variable. Three row bits give eight rows. One column bit gives two columns. Again the product is sixteen, covering the same assignments.

Recompute the address · 1:22

Our assignment now has row bits one-zero-one, worth four plus zero plus one: index five. Column bit one gives index one. The cell still contains one. Its address changed because the variable groups changed; its Boolean answer did not.

Your turn: track a zero · 1:40

Use assignment one-one-one-zero and the two-by-eight layout. What is its new row index? What is its new column index? You already know the rule returns zero for this assignment. Find where that zero belongs now.

Prediction pause: 10 seconds.

Answer: row one, column six · 2:05

A gives row one. B C D is one-one-zero, worth four plus two plus zero: column six. The entry is zero. The old four-by-four address was row three, column two. Both addresses refer to the same complete input assignment.

What is preserved? · 2:20

To check a repartition, compare answers for matching assignments, not matching physical positions. Every assignment must retain its output. With four variables we can check all sixteen directly. A new shape does not reduce the number of explicit answers.

Know what this lesson establishes · 2:35

These rectangles belong to the repository row-column layout convention. The paper tensor construction we studied was square. The examples here demonstrate addressing and representation. They do not, by themselves, show a speed advantage or a smaller total output.

You can now follow the whole path · 2:51

You can now read a Boolean rule, understand its matrix entries, construct larger paper examples, and track repository assignments through different layouts. When a grid looks confusing, begin with its variable groups and ordering. They are the key to making every cell meaningful.