This hypothetical, no-stakes worksheet uses Bicycle’s Continental Rummy rules. First check whether the proposed groups contain exactly fifteen cards; actual card identities require a separate check.
Three exact partitions
Start with the five-groups-of-three pattern. Five groups multiplied by three cards gives 5 × 3 = 15.
Next, test three groups of four plus one group of three. The calculation is (3 × 4) + 3 = 12 + 3 = 15. Keep the final three outside the multiplication; otherwise the calculation counts four equal groups instead.
The third pattern is one group of five, one group of four, and two groups of three. Its calculation is 5 + 4 + (2 × 3) = 5 + 4 + 6 = 15. Check that the final pair of threes is included exactly once.
A tempting count that fails
Consider the proposed arrangement 4 + 4 + 4 + 4. It looks balanced, but the calculation is 4 × 4 = 16. Therefore it cannot partition fifteen cards exactly: it is one card over. Removing one card would produce fifteen, but it would no longer be four complete groups of four.
| Pattern | Calculation | Total |
|---|---|---|
| Five groups of three | 5 × 3 | 15 |
| Three groups of four, one of three | 3 × 4 + 3 | 15 |
| Five, four, and two threes | 5 + 4 + 3 + 3 | 15 |
| Four groups of four | 4 × 4 | 16 |
What the arithmetic does not prove
A total of fifteen proves only that the proposed group sizes fit the card count. It does not establish suits, ranks, sequence order, or whether the actual cards form valid sequences. The displayed patterns correspond to the source’s specified sequence sizes, not permission to use arbitrary groups. Mark each paper card once while allocating it to a group. A card counted in two groups can make the arithmetic appear correct even though the physical allocation is impossible.
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