In the game of Sudoku, a 9 × 9 grid is to be filled in with the digits 1 – 9, such that in each column, each row, and each 3 × 3 sub-grid, each of the nine digits only appears once. There are other aspects to the game that make it easier or harder as a puzzle, but it’s the challenge of the filled-in grid that appeals to me.

In this series of images, I concentrated on a 4 × 4 grid and I used shapes instead of the digits 1 – 4. I retained the requirement that each element of the four-character alphabet appear once and only once in each row, column, and in each of the four 2 × 2 sub-grids. In addition, I added an element of layering: Each finished image is a composition of four layers, with each layer being its own solved Sudoku grid. The alphabets of image elements varied across the layers, but were chosen to be consistent across the image.

Consider Sudoku 4b, shown in Figure 1. The finished image is shown in the left panel and the right panel includes outlines to show how the 4 × 4 grid to separates the 16 squares.

Figure 1: Sudoku 4b, without and with grid outline.

Studying the cells, it may seem that the elemental images are based on quarter- and half-circles. That is indeed the case and the first layer and its alphabet are shown in Figure 2.

Figure 2: First layer, layer with grid, and corner showing alphabet.

For each layer, the alphabet of elemental images was composed of rotations of a primary shape, as suggested in Figure 2. The shapes and colors of the elements in all four alphabets were chosen to be complementary and so that they would effectively show their own colors and a wide variety of colors when merged. The primary shapes are shown in Figure 3.

Figure 3: Primary shapes for Sudoku 4b image alphabets.

In Sudoku 4a, the alphabets are based on rectangles. They are variously: solid and dark, solid and light, split light/dark, or split dark/light. The four primary images, one layer’s alphabet, and the finished image are shown in Figure 4.

Figure 4: Sudoku 4a: primary images, single layer’s alphabet, and final image.

Sudoku 4c is structurally similar to 4b. In this case, the primary images are triangular regions that cover (positive space) or leave uncovered (negative space) one-quarter or one-half of the grid square. The primary images and final piece are shown in Figure 5.

Figure 5: Sudoku 4c primary images and final image.

The final image in the series, Sudoku 4d, employed Truchet tiles for its basis. Each tile has two quarter-circle bands, which can be in either a top/right or top/left orientation. The orientation, along with the dark/light coloring (dark bands on a light background or light bands on a dark background) formed one of the four dimensions. The next two were the hues of the light regions and the hues of the dark regions, each with four possibilities. Finally, in each tile, the thick bands have thin bands laid on top of them. There are four choices for the orientations of the thin bands: none, parallel to the thick bands, crossing the thick bands, and four thin bands, two parallel and two crossing. A detail showing each of the four dimensions and the final image are shown in Figure 6.

Figure 6: Sudoku 4d detail and final image.

This series of four images explored the concept of filling a 4 × 4 grid in a Sudoku-like manner, except using geometric shapes and colors to create the alphabet of elements. In addition, layering of different grids was employed to highlight the interplay between structures.