Sum Select: find every answer without counting one twice.

Use a largest-tile ledger to find all nine solutions on one fixed board, then retry pairs, triples, and a four-tile answer.

H5 Free Fun · Published October 19, 2026 · 7 min read

Finding a group that reaches the target answers one question. Knowing that you have found every group answers another. A few successful guesses cannot establish completeness, and writing the same tiles in a different order does not create another answer.

This lesson gives you a small search ledger you can finish by hand. We will assign every answer to its largest selected value, search each branch once, and verify the complete list on a fixed Sum Select board. You can then retry the same board to compare answers of different lengths.

Load nine different values and a target of 10

Choose Practice this example in the side panel, or Load all-answers example below the game. Loading replaces the current session with the board below and target 10. Rows A–C run top to bottom; columns 1–3 run left to right. A1 holds 1, A2 holds 2, and C3 holds 9.

1 2 3
4 5 6
7 8 9

Each physical tile can be selected once. Selecting it again removes it. Any nonempty group with the exact total is accepted; the game does not require three tiles or the shortest answer. The target is compatible with the usual three-tile construction: 1 + 2 + 7 = 10, for example.

Give each answer exactly one home

Every nonempty selection on this board has one largest value because all nine values are different. Use that value as the name of its branch. If the largest value is 7, include 7 and search only values below 7 for the remaining 3. Do not include 8 or 9: they would make it a different branch.

Within a branch, write the smaller values in increasing order. The group 1, 2, 7 and the order 7, 1, 2 use the same three tiles, so record them once. A tile cannot partner with itself: a remainder of 4 cannot be filled by 2 + 2 here because there is only one tile showing 2.

This partitions the answers. No answer is lost, since every answer has a largest value. No answer belongs to two branches, since its largest value cannot be both 7 and 8. You must still finish the search inside each branch; choosing a branch label is not a completeness proof by itself.

Work downward through the largest values

• Largest 9: the remainder is 1. The only group is 1, giving 1 + 9.
• Largest 8: the remainder is 2. Use 2. You cannot use the single 1 twice.
• Largest 7: the remainder is 3. Use 3, or the two different values 1 + 2.
• Largest 6: the remainder is 4. Use 4, or 1 + 3. Two 2 tiles are unavailable, and the three smallest values already total 6.
• Largest 5: the remainder is 5, using only 1–4. Use 1 + 4 or 2 + 3. There is no smaller single 5, and any three distinct smaller values total at least 1 + 2 + 3 = 6.
• Largest 4: the remainder is 6, using only 1, 2, and 3. Their entire total is 6, so all three are needed.

Stop below 4. If the largest selected value is at most 3, all available values together total only 1 + 2 + 3 = 6. No subset of them can reach 10. Positivity justifies this upper-bound check: omitting a positive value cannot increase the sum.

For the small remainders 1, 2, and 3, the listed singletons and distinct-value pairs exhaust the possibilities. Three selected values would already total at least 6. For remainder 4 the same bound excludes triples. For remainders 5 and 6 the final two branches above check the remaining possibilities explicitly.

Read the complete answer ledger

There are exactly nine selections: four pairs, four triples, and one four-tile group. The coordinates below identify the physical buttons. Values are written in increasing order to make duplicate records easy to spot.

1. A1 + C3: 1 + 9 = 10.
2. A2 + C2: 2 + 8 = 10.
3. A3 + C1: 3 + 7 = 10.
4. A1 + A2 + C1: 1 + 2 + 7 = 10.
5. B1 + B3: 4 + 6 = 10.
6. A1 + A3 + B3: 1 + 3 + 6 = 10.
7. A1 + B1 + B2: 1 + 4 + 5 = 10.
8. A2 + A3 + B2: 2 + 3 + 5 = 10.
9. A1 + A2 + A3 + B1: 1 + 2 + 3 + 4 = 10.

No tile equals 10, while four two-tile groups do reach it. Two is therefore the minimum selection length. The four-tile answer is still valid. “All answers” and “all shortest answers” are different lists.

Compare three answers on the same live board

On the loaded example, select A1 and C3 for 1 + 9, then choose Check sum. The game accepts 10 and keeps the answer visible. Choose Retry this board to clear the selection while keeping the same nine values and target.

Next select A1, A2, and C1 for 1 + 2 + 7, then check. Retry again and select A1, A2, A3, and B1 for 1 + 2 + 3 + 4. All three lengths work. You can repeat this routine for the other six records in the ledger.

The game verifies each submitted total; it does not remember which answers you have already found or declare your list complete. Keep your ledger on paper. A correct practice answer does not advance to a random next round. New puzzle returns to the ordinary ten-round session; refreshing this fixed practice entry reloads the example with an empty selection.

Two checks for your search

When this ledger needs a different convention

The method here uses distinct values. On another board with repeated values, two different physical tiles can tie for largest. To keep one home per selection, order the tiles by value and then by coordinate, and assign the selection to its last tile in that fixed order. Search only earlier positions. Do not merge two equal-valued tiles into one resource.

Exhaustive listing is useful when you want a proof of completeness. It is more work than finding one valid group, which is all a normal round asks for. The current positive 1–9 values make remainder and upper-bound pruning valid; do not carry those shortcuts unchanged to a puzzle with negative numbers.

The printed board and all 512 subsets of its nine positions were independently evaluated to verify the nine records, their lengths, and the branch counts. That is a check of this fixed example, not a promise that every random round has nine answers or that a particular method is always fastest.