The number of crosspoints needed for 10 lines in a crosspoint switch, which is…

Concept:

A crosspoint switch (a crossbar) lays its lines out as the rows and the columns of a matrix. At every row-column intersection sits one switching element, called a crosspoint, and closing that element joins the line on that row to the line on that column. The number of crosspoints is therefore simply the number of intersections the design actually has to build.

A crosspoint switch that serves a set of lines uses that same set as both its rows and its columns, so its matrix is symmetric about the diagonal; such a layout is called a triangular or two-way matrix. Two stated conditions then decide how much of that matrix actually has to be built, and each one removes a specific part of it. Full duplex is the condition that collapses the symmetry: a connected pair exchanges traffic in both directions over the same physical circuit, so one crosspoint already carries the whole conversation between the two lines, and the mirrored intersections (i, j) and (j, i) are the same crosspoint counted twice rather than two separate switching elements. Two separate elements would be needed only if each direction of travel had to be switched on its own path, which is exactly what full duplex over one circuit rules out. Excluding self-connections removes the diagonal, where a line would meet itself.

What survives is exactly one crosspoint for each unordered pair of two distinct lines. For n lines that count is C(n, 2) = n(n − 1)/2. This is the standard textbook result for a crossbar serving n lines when the lines are full duplex and self-connections are excluded: the complete n × n matrix reduces to just the crosspoints lying above the diagonal.

Application:

1. Lay the 10 lines out as both the rows and the columns: 10 × 10 = 100 intersections in the complete matrix.

2. Remove the 10 diagonal intersections, because self-connections are excluded: 100 − 10 = 90 intersections remain.

3. Fold the matrix along its diagonal: full duplex makes (i, j) and (j, i) one and the same physical crosspoint, so each pair of lines is counted once instead of twice: 90 ÷ 2 = 45.

Cross-check:

Counting the pairs directly gives the same figure: C(10, 2) = (10 × 9)/2 = 45.

Contrast:

Each of the other counting conventions honours at most one of the two stated conditions: the 10 × 10 count applies neither, the 10 × 9 count drops the self-connections but still pays for each direction of travel separately, and the (10 × 10)/2 count halves the matrix but keeps the diagonal a line would use on itself. The (10 × 9)/2 convention is the only one that honours both stated conditions at once — full-duplex lines sharing one circuit, and no self-connections — so the switch needs 45 crosspoints.