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Bonus Chapter 28›Lesson 28.4
Constraint puzzle studio · cards, lines, and controlled change

Arrange and Move Cards to Control Line Sums

Coordinate many rows, columns, and diagonals at once. Then make one carefully chosen paper-card move so that all six row and column totals agree.

13 playing cardstarget 21flexible aceone-card movedifference tracking
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Two tasks: arrange cards, then repair six totals

  1. Playing cards, Missions 1–6: use A, 2–10, J, Q, and K once each. Build an arrangement with at least 12 scored lines, all totaling 21. First learn to read a line, inspect a starting arrangement, and study how a flexible ace helps. Then check a complete example and build your own board.
  2. Paper digit cards, Missions 7–8: move exactly one digit card to cover a card in another cell. You may rotate a moved 9 into a 6. Make all three row totals and all three column totals equal 31.

In both puzzles, one card can affect several lines. Before changing it, predict which lines will change and by how much.

Two puzzles, one big idea

A single card can belong to several lines. Moving or changing it updates every one of those lines at once.

line total = sum of every card on that row, column, or diagonal
best move = a small change that repairs several conditions together

The playing-card rules

J, Q, and K each count as 10. The ace may count as 1 or 11, chosen separately for each line. For example, J + A + Q = 21 uses A = 1, while 10 + A = 21 uses A = 11. It is still the same card.

Count each whole row, column, or 45° diagonal through cell centers that contains at least two cards. Add every card on that line, even if there are gaps between cards. Count the line once; do not count smaller groups within it as extra lines.

1
Read every condition

Decode the line-sum card puzzle

Not complete
Face cardsJ=Q=K=10
Scored lineAt least two cards share a row, column, or 45° diagonal.
Construction goalAt least 12 scored lines, with every scored line totaling 21.
Whole-board checkEvery card is used once; every qualifying line is counted.
Do not count a one-card line. A card standing alone in its row or diagonal does not create a scored total under these rules.
Checkpoint
2
Read one line at a time

Scan rows, columns, and diagonals

Not complete
Learn how to read a scored line. This board is a supplied all-21 example. Select a row, column, or diagonal to highlight its cards and equation. In Mission 5 you will audit the same board as a whole; next, Mission 3 gives you a different starting arrangement to investigate.

A completed example: read one line

Select any scored line. The highlighted cards and exact equation will appear below.

Cards on line—
Ace value used—
Line total—
Target?—
Checkpoint
3
Inspect the board you will improve

Count the starting arrangement’s line totals

Not complete
Inspect a different starting arrangement. This is the board that opens in your Mission 6 editor. Reveal its lines to see what needs improving. Every total below is calculated from the cards shown, using the same rules as your editor. The checkpoint asks about the whole arrangement, regardless of how many lines you have revealed.

Inspect each row, column, and diagonal

Reveal the next line, add its highlighted cards, and compare with 21. The numbered totals and equations follow the same order as the live board’s line list.

Lines inspected so far0 / 13
Inspected lines totaling 210
Inspected lines with other totals0
Share of inspected lines totaling 210%
Review inspected line equations
Check the non-target lines too. J + Q = 10 + 10 = 20, and the diagonal 2 + 8 + J also totals 20. A line that misses 21 still counts as a scored line.
Checkpoint
4
Plan a value separately for each line

Use one ace in several lines

Not complete
Prepare to improve the starting board. Look back at the completed example in Mission 2. Its central ace helps several lines reach 21 because you may choose its value separately on each line. Work out the needed values below before auditing the whole example in Mission 5.

Find the value a line needs

needed ace = 21 − sum of the other cards

If the other cards total 20, use A = 1. If they total 10, use A = 11. If neither choice makes 21, that line needs a different arrangement.

Keep the card; change how you count it

A = 1 or 11 on each line

The same ace can help several crossing lines. Choosing A = 1 for a row does not force A = 1 for its column or diagonals.

Center row
J+A+Q=10+1+10=21
Center column
10+A=10+11=21
Center diagonal
4+A+6=4+11+6=21
Check every line through the ace. A value that works for one line does not automatically fix the others. Choose 1 or 11 for each line and verify its sum.
Checkpoint
5
Check the whole worked example

Check a complete all-21 arrangement

Not complete
Return to the completed example from Mission 2. Earlier you learned to read individual lines. Now inspect every qualifying line, so you can justify that the whole arrangement meets the goal. The board is already complete; Next line changes only which lines you have inspected.

Inspect one scored line at a time

About the complete arrangement

Total scored lines12
Cards used13

Your inspection progress

Lines inspected so far0 / 12
Inspected lines totaling 210
What does a complete check establish? All 13 cards are used once. The arrangement has 12 qualifying rows, columns, and diagonals, and every one totals 21. This meets the construction goal. The questions below concern this whole arrangement, not just the lines inspected so far.
Checkpoint
6
Independent construction

Move cards and optimize the 7×7 board

Not complete
Your playing-card construction. Begin with the starting arrangement you inspected in Mission 3. Move or swap cards to make every scored line total 21, with at least 12 scored lines. Use the line list to find what still needs work.

Your board’s line totals

Click a card, then click another cell to move or swap it. The checker scores every row, column, and 45° diagonal containing at least two cards. A may be 1 or 11 separately on each line.

Scored lines—
21 totals—
Other totals—
Target rate—
Select a card and a destination. The coach seeks more 21-lines; its suggestion need not make every line equal 21.
Mission goal

Produce a board with at least 12 scored lines, all totaling 21. Try your own moves and validate the result. Then, if you need help, open a separate worked arrangement in solution review. An optional further challenge is to seek even more lines totaling 21; you do not need to prove a maximum to complete this mission.

7
Begin the separate paper-card puzzle

Read the six starting line totals

Not complete
New cards and a new goal. These are paper digit cards, not playing cards. Digits next to each other in a cell form a numeral. First inspect the six starting totals; in Mission 8, make them all 31 with exactly one card move. Only rows and columns count in this puzzle.

Initial row and column totals

This is a separate puzzle with 14 paper digit cards in nine cells. Read the digits in each cell from left to right as one numeral: two cards showing 1 and 9 mean 19, not 1+9. A cell emptied by the move has value 0.

The middle row and middle column already total 31. A successful move should repair the four outer lines without damaging those two.
Checkpoint
8
One card, four repaired lines

Move, rotate, cover, and verify

Not complete
Your one-move attempt. Compare each starting total with 31. Find one move that changes the four outer lines by the amounts they need, while preserving the middle row and column. The worked explanation below is available if you get stuck.

Move controls

  1. Click the paper card you will move.
  2. If it is a 9, choose whether to rotate it into a 6.
  3. Click a card in another cell to cover. The moved card leaves its original cell and replaces the covered digit’s visible face; the covered card contributes nothing separately. Keep all other digit positions unchanged. The preview updates immediately.
Worked explanation: one move repairs four lines

The bottom row and left column each total 40, so each must lose 9. Their crossing cell contains a single 9. Removing it leaves that cell empty, with value 0.

The top row and right column each total 34, so each must lose 3. At their crossing, turn 19 into 16: rotate the removed 9 to make 6, then cover the 9 in that top-right cell. The middle row and column stay at 31. All six totals are now 31.

Exact one-move audit

The added checker tests every ordered original-card and destination-card choice in different cells, including the extra rotated-face option for a moved 9.

Legal move cases—
Equalizing moves—
Common total—
Only one successful move?—
Mission goal

Make all three rows and all three columns equal to 31 with exactly one card move. Explain which lines your move changes and why the other lines stay correct. Checking every possible move is optional.

9
Independent review

Line-sum card workshop

Not complete
10
Final check

Exit ticket and certificate

Not complete
Bonus Chapter 28 · Lesson 28.4

Line-Sum Card Constraint Solver

This certifies that

Learner

coordinated card values across rows, columns, and diagonals, then used one controlled move to equalize six line totals.

Awarded after all ten missions and the exit ticket are complete.