01
Reconstruct all twelve cells before comparing sizes
The first row is zero, seven, zero, zero. The second row is zero, zero, zero, five. The third row is two, zero, zero, nine. The coordinate list contains (1,2,7), (2,4,5), (3,1,2) and (3,4,9). Reconstructing the rows checks that each listed value lands in the intended cell and that all eight other cells use the declared zero default. The example has no repeated coordinates. If duplicates were allowed, a separate rule would need to say whether to combine them, reject them or let one entry replace another.
02
Count coordinates as well as stored values
The dense representation uses twelve value slots. Four triples use three slots each, also twelve slots, so this example is a tie under the stated model. With k nonzero entries, the coordinate list uses three times k slots. It is smaller than the twelve-slot dense grid only when k is less than four: zero through three entries. Five entries would require fifteen slots. This comparison does not predict actual memory use because real formats may use different index widths, headers, alignment and compression. Keep those implementation details outside the simple model until they are explicitly measured.
03
Record the assumptions with the result
A reusable worksheet records row and column counts, index origin, coordinate order, default value, duplicate policy and the number of occupied positions. It then records the counting model, dense total, coordinate-list total and comparison result. For this grid the totals are twelve and twelve, with no storage advantage in the equal-size-slot model. A later exercise can change the occupancy while retaining the dimensions to find the threshold directly. Preserve the reconstructed grid alongside the list so another reader can check both the representation and the arithmetic without relying on a visual impression of sparseness.