Sum Select: equal numbers can make different answers.
Separate value patterns from physical selections, fill a six-row count ledger, and explain all 18 answers on one repeated-value board.
Two tiles both show 3. Does choosing them give two ways to make 6, or just one? The answer depends on what you are counting. Their two positions form one selected group. Clicking the left 3 before the right 3 does not create a second group.
Repeated values also create a different effect: replacing a selected tile with another tile of the same value changes the selected positions without changing the sum. This lesson separates those two ideas. You will count six value patterns and eighteen physical selections on one small board, then check why the list is complete.
Keep the positions visible
Use this constructed paper board with target 6. Rows A–C run top to bottom, and columns 1–3 run left to right. Write coordinates beside your answers so the equal numbers remain separate resources.
1 1 2 2 3 3 4 4 5
A1 and A2 are the two 1 tiles; A3 and B1 are the two 2 tiles; B2 and B3 are the two 3 tiles; C1 and C2 are the two 4 tiles. C3 is the only 5. Each position can be selected at most once.
This is a paper exercise, not a layout available from a loading button. The live game's rule accepts any nonempty group with the exact target, and selecting a tile again removes it. Our target is compatible with the ordinary generator's three-position construction: A1, A3, and B2 give 1 + 2 + 3 = 6. The example does not claim every random round has these values or this many answers.
Name the object you are counting
A value pattern records the selected values in increasing order, keeping repeats. A physical selection records the selected coordinates without regard to click order.
A1 + C3 and A2 + C3 are different selections, both with value pattern 1 + 5 and total 6. There are two selections for that pattern because either 1 can accompany the single 5.
By contrast, B2 + B3 and B3 + B2 select exactly the same two coordinates. Both use the two available 3 tiles. The pattern 3 + 3 contributes one selection, not two. Repeated clicking never makes a tile available twice.
Find every value pattern before counting positions
Group the patterns by the number of selected tiles. All values are positive, so small and large selection sizes can be bounded directly.
- One tile: none, because there is no 6.
- Two tiles: the unordered positive pairs are 1 + 5, 2 + 4, and 3 + 3. Each is available.
- Three tiles: the nondecreasing positive triples totaling 6 are 1 + 1 + 4, 1 + 2 + 3, and 2 + 2 + 2. The last needs three 2 tiles; this board has only two. Keep the first two patterns.
- Four tiles: the four smallest available values are 1, 1, 2, and 2, already totaling 6. Replacing any of them with a larger value exceeds the target, so 1 + 1 + 2 + 2 is the only pattern.
- Five or more: the five smallest values total 1 + 1 + 2 + 2 + 3 = 9. No larger selection can total 6.
The six surviving patterns are therefore complete. An unavailable pattern such as 2 + 2 + 2 is a resource error, even though its arithmetic is correct.
Fill the count ledger
For each pattern, count coordinate sets rather than clicking sequences. Enter your six predictions below, or write them on paper. These optional fields are a scratchpad: they do not submit, grade, or save your work. Reveal the answer ledger when you are ready to compare. Closing or refreshing this page is not a way to keep a record.
Reveal all six counts and one selection for each
| Values | One selection | Count |
|---|---|---|
| 1 + 5 | A1 + C3 | 2 |
| 2 + 4 | A3 + C1 | 4 |
| 3 + 3 | B2 + B3 | 1 |
| 1 + 1 + 4 | A1 + A2 + C1 | 2 |
| 1 + 2 + 3 | A1 + A3 + B2 | 8 |
| 1 + 1 + 2 + 2 | A1 + A2 + A3 + B1 | 1 |
Add the counts: 2 + 4 + 1 + 2 + 8 + 1 = 18 physical selections. Grouped by length, there are seven pairs, ten triples, and one four-tile selection. There are six value patterns, not eighteen.
Multiply choices, but do not multiply order
For 2 + 4, choose one of two 2 tiles and one of two 4 tiles. The choices are independent: 2 × 2 = 4. List them to check: A3 + C1, A3 + C2, B1 + C1, and B1 + C2.
For 1 + 2 + 3, each of three value groups offers two choices, so there are 2 × 2 × 2 = 8 selections. Do not multiply by six possible click orders. A1, A3, and B2 are the same group however you click them.
When a pattern needs both copies of a value, there is just one choice from that value group. For 1 + 1 + 4, both 1 tiles are required, while either 4 works: 1 × 2 = 2. For 1 + 1 + 2 + 2, both 1 tiles and both 2 tiles are required: 1 × 1 = 1.
On another board with three copies of 2, choosing two would give three coordinate pairs: first/second, first/third, and second/third. It would not give six. Count subsets within each value group, then multiply across different values. A request for more copies than the board holds contributes zero.
Remove one tile and test your reasoning
For a paper variation, cross out A2, leaving only one 1. Keep target 6 and all other tiles. Before revealing the answer, decide which patterns disappear and which merely lose choices.
How many selections remain without A2?
1 + 5 now has one; 2 + 4 still has four; 3 + 3 still has one; 1 + 1 + 4 disappears; 1 + 2 + 3 falls to 1 × 2 × 2 = 4; 1 + 1 + 2 + 2 disappears. The total is 1 + 4 + 1 + 0 + 4 + 0 = 10. Exactly eight of the original eighteen selections used A2. Removing one physical tile does not remove every pattern containing its value.
Use the right level of detail
A value-pattern ledger is compact and useful for completeness checks. A coordinate ledger tells you which actual buttons to select. Translate between them before declaring two answers equal or different.
A normal Sum Select round asks for one valid selection, not every answer or the fewest tiles. Exhaustive counting is optional practice. The bounds here use positive integer values and this exact stock of tiles; a puzzle with negative numbers, reusable tiles, or a rule that makes order matter needs a different analysis. No probability of winning follows from these counts without a defined random selection method.