A named construction · An inspectable example

Follow a rule.
Explore a phase.

Choose a grid size, move through its phases, and select a cell to see the arithmetic behind its value.

This explorer uses the published TLF seed with explicit, one-based indexing. It is one named example, rather than a definition of the whole Naglich Squares family.

Your finite example

Hide the numbers to follow the color patterns, or emphasize a value group. Every cell keeps its accessible value and arithmetic; dimmed cells remain selectable. Tab into a grid, then use arrow keys to choose a cell. Home and End move along a row. Phase buttons wrap from N back to 1.

Value distribution · Grid A

Counts of raw represented values 1 through N in Grid A, including dimmed cells. These are different from nine digital-root frequencies or three-family counts.

The rule stays visible.

A_TLF(r,c,k,N) = (((r − 1) × wrapN(c − 1,N) + (k − 1)) mod N) + 1

wrapN(x,N) = ((x − 1) mod N) + 1

N: The number of rows and columns; this explorer accepts integers from 2 to 16.
r, c: Row and column indices, each from 1 to N.
k: The phase for grid A, from 1 to N. Grid B uses the same rule with phase j.
mod: The nonnegative remainder: always from 0 to N − 1 here.

Indexing and limits of this example

The wrap is applied to c − 1, not directly to c. At column 1, wrapN(0,N) equals N. All coordinates and phases are one-based.

Comparisons count equal values at matching coordinates in the two grids currently shown. They do not establish a theorem, a physical interpretation, or a relationship across the full family.

The phase-stack viewer uses the same named TLF seed. Family rules are presented separately so their orientations remain explicit.

Compare the eight family rules →Explore a phase stack →

Compare distinct instruments

The legacy heatmap adds playback, palettes, a histogram and sequence controls. Square Grids compares product and affine generators on dual boards; export behavior has separate browser and sandbox limits.