Build Latin squares with trailing-zero place values
A 4 can be 4, 40, or 400. But it is still the same base digit. Can you give every digit its own place—and make all the totals agree?
Checkpoints: 0 of 4.
One grid. Two jobs.
A number in a cell carries two pieces of information: its base digit and how many zeros follow it.
Keep the digit. Change its size.
On the 4 × 4 board, each row and each column must contain 1, 2, 3, and 4 exactly once. On the 6 × 6 board, use 1 through 6.
3, 30, and 300 are allowed versions of base digit 3. The number 303 is not. Zero by itself is not a base digit.
Add the actual values—not just the base digits. Row targets are at the right; column targets are below the grid.
An arrangement in which each allowed symbol appears once in every row and column. This puzzle has no diagonal rule and no smaller-box rule.
“Trailing” means “at the end.” Adding a zero to the right multiplies the value by 10. Choosing no zeros leaves the digit unchanged.
How big should one digit be?
This magnifier explores values. A value belongs in a puzzle cell only when the row and column targets can accommodate it.
A quick rule check
One 4-cell row contains 10, 20, 100, 4. Does it use each base digit 1–4 exactly once?
Let the totals help you place the digits
Separate the two layers in your thinking, but use them together. A row total can tell you which values are possible before you know their order.
original, Figures 28–22 and 28–23. The book supplies the task and one finished grid. The staged deductions and choice explanations here are teaching additions; a displayed choice is not automatically a forced move.
Your two-layer workshop
Pick a cell, choose a base digit, then choose its zeros. Watch the selected row and column—not just the cell you changed.
Make all eight totals agree
The original board starts empty. Every target is fixed.
The target is bold. The smaller number is your current sum; an asterisk means some values are still undecided. Tap a total to inspect that line.
Inspect row 1
Edit one cell
R1 · C1Coordinates & keyboard controls
On a focused grid cell, use arrow keys to move. Press 1–4 for a base digit, + or − to change zeros, ? to leave the zero choice undecided, and Delete or Backspace to erase. Tab reaches the editor controls.
A digit entered in an empty cell starts with 0 trailing zeros. Changing an existing base digit keeps its current zero choice (including “?”). Use “?” when you want to plan the base arrangement before deciding the values. You can undo your own edits and erasures. Solution review preserves your grid.
Study the book’s 4 × 4 answer or start with its base digits
Example 9 · Figure 28–23.
The checker accepts every valid answer, not just this grid. Keep both the Latin rule and the printed totals.
Loads the book’s digit arrangement with all zero choices marked ?. This is a scaffold, not the empty original puzzle.
Both buttons replace the active grid. Undo restores your work.
Checkpoints record that a valid grid was checked, not that it was completed without help. The two boards keep separate work. A green sum is only one check: all base-digit and total checks must pass together.
Paper puzzle workspaces
| C1 | C2 | C3 | C4 | Target | |
|---|---|---|---|---|---|
| R1 | 343 | ||||
| R2 | 172 | ||||
| R3 | 190 | ||||
| R4 | 424 | ||||
| ↓ | 190 | 334 | 460 | 145 |
| C1 | C2 | C3 | C4 | C5 | C6 | Target | |
|---|---|---|---|---|---|---|---|
| R1 | 1443 | ||||||
| R2 | 237 | ||||||
| R3 | 930 | ||||||
| R4 | 273 | ||||||
| R5 | 1326 | ||||||
| R6 | 849 | ||||||
| ↓ | 1317 | 615 | 291 | 507 | 984 | 1344 |
Carry carefully. Coordinate both directions.
Place-value groups can need regrouping. And a change that preserves the Latin rule can still break the totals.
Thirteen ones are allowed in a sum
The worked example’s fourth row has these six values:
Do not read a target’s last digit as the whole ones contribution. A target ending in 3 may come from 3 ones—or from 13 ones with a carry.
A Latin square is not enough
Start with the small worked solution. Now swap its first two whole rows, but leave the printed targets where they are.
Digit positions and zero choices constrain each other. You may need to revise either one.
One sum narrows the choices. All lines decide.
A set of values can match one target without fitting the entire grid. Explore that difference, then ask how many complete answers exist.
What values can make this total?
Use each base digit once. Try every possible number of trailing zeros that fits the selected total. The order of the values is not decided here.
Can two different sets make the same total?
This laboratory ignores cell positions and all crossing lines. It does not change your workshop. Different orders of the same values are grouped as one set.
Is the worked solution the only answer?
Search from the empty original board, preserving all printed row and column targets. This does not use or change your workshop entries.
Replaces that board’s entries. Undo restores them.
Why the search is finite—and what it proves
Every cell is positive and is part of a row and a column total. Its value cannot exceed the smaller of those totals. In the 4 × 4 puzzle, 1,000 is already too large everywhere. In the 6 × 6 puzzle, some cells can initially allow 1,000; values with four trailing zeros are too large everywhere.
The search lists every value set for each line, then every order of each set. It removes a line candidate when a crossing line cannot support one of its entries. It then tries the remaining candidates systematically, undoing each trial to explore the next branch.
A finished search covers all complete grids under these rules. A search stopped by its time or result limit is reported as unfinished—not as a uniqueness or impossibility proof. The count, reasoning, and search program are additions for this lesson, not claims made by the book.
Explain your thinking
Check the ideas behind the grids. A finished answer should survive more than a quick glance.
Four checkpoints complete
You have checked both grids and explained the key rules. Keep exploring—there may be more to discover than one finished answer.
Learning notes, precise rules & teaching additions
original task: put 1–4 in a 4 × 4 grid, or 1–6 in a 6 × 6 grid, so each digit occurs in each row and column. Then append zeros so the values match the surrounding totals. There are exactly as many cells in a line as allowed digits, so each digit occurs exactly once in that line. The printed constructions include unscaled digits, so “append zeros” includes appending none. The worked example does not impose diagonal conditions, subgrid conditions, or a shared number of zeros for all appearances of a base digit.
Exercise notes values: the following tables transcribe the two printed answers.
Example 9 · Figure 28–23
| 40 | 200 | 100 | 3 |
| 30 | 100 | 40 | 2 |
| 100 | 30 | 20 | 40 |
| 20 | 4 | 300 | 100 |
Rows: 343, 172, 190, 424.
Columns: 190, 334, 460, 145.
Practice 8 · Reference item 8
| 300 | 500 | 40 | 2 | 600 | 1 |
| 2 | 60 | 5 | 100 | 30 | 40 |
| 600 | 30 | 10 | 40 | 50 | 200 |
| 5 | 1 | 200 | 60 | 4 | 3 |
| 400 | 20 | 6 | 300 | 100 | 500 |
| 10 | 4 | 30 | 5 | 200 | 600 |
Rows: 1443, 237, 930, 273, 1326, 849.
Columns: 1317, 615, 291, 507, 984, 1344.
What the book does not supply: it gives a finished grid for each task, but not the deductions in this guide, a recommended solving algorithm, a count of all answers, or a uniqueness proof. All step-by-step explanations, the magnifier, carry demonstration, row-swap experiment, interactive hints, value-set laboratory, whole-grid audit, and exit questions are teaching additions.
A useful distinction: we separate base digits from powers of ten in the editor. That is not a claim that an arbitrary Latin square can always be completed by adding zeros. The solver keeps both kinds of constraint active.
Bounds, not extra puzzle rules: each positive cell value is at most the smaller of its row and column targets. Thus 0–2 zeros cover all possible 4 × 4 entries. For the 6 × 6 grid the search also includes three-zero entries wherever they fit (only 1,000 can be small enough). Four zeros cannot fit either puzzle. The magnifier is unrestricted by a particular cell. Disabled editor choices are explained by the current cell’s target bound.
Checking: an entry must consist of a single allowed digit followed only by zeros. A row and column may each contain that base digit once, even when the two values use different powers of ten. Unknown zero choices are not treated as zero-valued cells. “Current sum” adds only decided entries; acceptance requires every cell to be decided, all Latin lines to pass, and every total to match. The checker does not compare entries with the reference answer.
Search and hints: A compatible-value hint gives one completion-preserving choice, not necessarily a forced one. All counts are computed by this page and are distinguished from original statements. An interrupted search is not used to prove uniqueness or impossibility.
Using the file: all diagrams, data, styles, and calculations are included. No external fonts, libraries, or network requests are required. The page works offline. Optional browser storage saves only this lesson’s boards, checkpoints, quiz choices, and reflection on this device. Blocking storage does not prevent play. The reset button removes this lesson’s saved work. Checkpoints do not distinguish independent work from hinted or revealed work.