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Puzzle Studio · Frequency Optimization

Lesson 27.7 — Select Grid Cells to Maximize Repeated Line-Sum Scores

Choose exactly twenty cells, add every row and column, then improve the product of a most-frequent line sum and its frequency. A strong construction balances three dials at once: the size of the repeated sum, its frequency, and the exact selection limit.

8 × 8 original grid20 selected cells16 line sumsbenchmark 240undo · redo · coach
Your progress and best legal score are saved in this browser.

Three boards to learn one scoring game

Your main challenge: select exactly 20 cells on the 8×8 board and try to make the same large sum appear in many rows and columns. First, use two smaller boards to learn the scoring and test strategies.

  1. 5×5 board — choose 10 cells: Missions 2–4 teach the line-sum ledger. Mission 2 asks for sum 10 on six lines, giving 60 points.
  2. 4×4 board — choose 6 cells: Mission 5 accepts any six-cell selection. Improving its score and proving the best possible score are optional investigations.
  3. 8×8 board — choose 20 cells: Mission 6 is your first attempt, Mission 7 explains a 240-point benchmark, and Mission 8 lets you improve and save attempts. You do not need to reach 240 to complete your construction checkpoints.

The allowed number of selected cells belongs to the board you are using. The scoring method stays the same.

The worked example scoring rule

Select the required cells. Add selected values in each row and each column. Find the line-sum value that occurs most often.

score = repeated line sum × number of appearances

An empty row or column receives line sum 0. On the 8×8 board, count all eight row sums and all eight column sums.

A small scoring example: suppose the row sums on a 3×3 board are 6, 6, 3 and its column sums are 6, 6, 3. Across these six lines, 6 appears four times and 3 appears twice. The most frequent sum is 6, so the score is 6 × 4 = 24.

Tie rule for every board on this page: first find the greatest frequency. If several sums share that frequency, use the one giving the largest product. For example, if 6 and 8 each appear twice and no sum appears more often, use 8 × 2 = 16. “Modal sums” means the sums tied for that greatest frequency.

The three dials

ValueA larger repeated sum can raise the product.
FrequencyA sum repeated on more lines can raise the product.
Cell budgetYou must use exactly the allowed number of cells.
Benchmark, not a proved maximum: the example scores 24 × 10 = 240. It is a useful target to compare with; this lesson does not establish the greatest possible score on the 8×8 board.
1

Decode the scoring system

Objects, line sums, and the exact cell budget
Not complete
ChooseExactly 20 of the 64 cells in the worked example grid.
AddCompute 8 row sums and 8 column sums.
GroupCount how often every sum value occurs.
ScoreMultiply the most frequent sum by its frequency.
2

Build the worked example’s 5 × 5 demonstration

Select ten cells and watch every line sum update
Not complete
Learn the scoring on a 5×5 board. Here the allowance is 10 cells. Watch how each chosen cell changes one row sum and one column sum. The score is provisional until exactly ten cells are selected; then try to make sum 10 appear on six lines.

Interactive Figure 27-14

Click a cell to place or remove a circle. The worked example demonstration circles ten values.

0 / 10 cells selected

Live score laboratory

selected0
largest frequency10
modal sums0
comparison score0
0
0 × 10
Provisional until exactly 10 cells are selected.

Mission: recreate the worked example demonstration, or find another ten-cell selection with repeated sum 10 appearing six times and score 60.

3

Explain why the demonstration scores 60

Read the row ledger, column ledger, and frequency table
Not complete
Explain the worked ten-cell selection. This ledger follows the board in Mission 2. Show the worked example on the guided board there if you changed it: the questions here name the fixed example, rather than your latest experiment.

Live line sums from Mission 2

The questions below refer to the worked example ten-cell demonstration. Use “Fill the guided board with the worked example” in Mission 2 to display that ledger again.

LineSelected valuesSum

From frequency to score

The repeated value is not the total of all selected cells. It is one row-or-column sum that occurs on several different lines.

10 appears 6 times, so 10 × 6 = 60
The same selected cell contributes to exactly two line sums: its row sum and its column sum.
4

Handle a frequency tie honestly

The worked example does not state what to do when several sums tie for most frequent
Not complete

Edit eight hypothetical line sums

Report every modal candidate

original gap: if two sums share the greatest frequency, the printed rule does not choose between their different products.
Added app convention: for all live scores, saved attempts, and the smaller-grid audit, this page uses the largest product among tied modal sums. It still displays every tied candidate.
5

Build a six-cell selection and investigate its best score

Added exact-search practice using the upper-left 4 × 4 original values
Not complete
New board: 4×4, with exactly six selections. Make a selection, inspect its score, and try a different arrangement. Any legal six-cell selection passes this checkpoint. The optional search below compares every possible six-cell selection to prove which score is greatest.

Select exactly six cells

0 / 6 cells selected

Your current six-cell experiment

selected0
modal sum0
frequency8
score0
Optional: prove the greatest six-cell score
The audit will check every 6-cell subset of the 16-cell board.
There are 8,008 different ways to choose six of these sixteen cells. The search checks them all; this total does not change when you edit your board.

Make a legal six-cell selection and inspect its row sums, column sums, and score. Improving it and running the exhaustive audit are optional challenges.

6

Make a first legal 20-cell plan

Explore the complete 8 × 8 original grid before studying its benchmark
Not complete
Your first main-board attempt: 8×8, exactly 20 cells. Use what you learned about repeated sums. Start with any legal selection, inspect its score, and then see the benchmark in Mission 7. There is no minimum score for this first construction.

original Figure 27-15

0 / 20 cells selected

Aim at one repeated target

selected0
lines at target0
largest frequency16
comparison score0
7

Reconstruct the worked example benchmark of 240

Ten of the sixteen line sums become 24
Not complete
Study a completed twenty-cell example. The slider reveals its chosen cells gradually. The ledger and frequency bars describe only the cells revealed so far. The full example scores 240; answer the checkpoint about that complete example, using Reveal all when you want to inspect it.

Reveal the twenty-cell example step by step

Ledger for the cells revealed so far

cells revealed0
most frequent sum(s) now0
greatest frequency now16
score for this revealed portion0

Reveal all 20 cells to see a complete legal selection.

8

Optimize freely and keep an attempt history

Match 240, exceed it, or explain what your next swap is trying to improve
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Full challenge board

0 / 20 cells selected

Live optimizer

current score0
personal best0
modal sum0
frequency16
0
0 × 16
Select exactly 20 cells to create a legal score.
One-swap coach: select exactly 20 cells first. The coach tests every remove-one/add-one exchange.

Saved attempts

No saved legal attempts yet.

Optimization reflection

This mission completes after you save one legal 20-cell attempt and identify a productive optimization goal. Reaching 240 is an optional challenge.

9

Repeated line-sum workshop

Eight objective checks · all eight correct complete the mission
Not complete
10

Exit ticket and certificate

Five correct answers unlock the studio certificate
Not complete