Problem
Fill the grid under row-column-region rules
A 5 by 5 table already holds nine numbers. Every row and every column must hold 1 to 5 once each. So must each bold five-cell region. Fill in all sixteen empty cells.
Reasoning · Grade 5-2 Number Arrangements
Your answer
Watch out The regions look like blocks, but they wind across the grid — mark them first.
How to solve
Strategy Eliminate Possibilities — Every empty cell has only five candidates, 1 to 5, so this is exactly the situation tool 3 was made for: cross off the numbers a cell is forbidden, and if only one survives, that cell is settled. Guessing a whole grid and repairing it would be slow and error-prone, because one wrong entry poisons a row, a column and a region at once. The trick is to hunt for the cell that is squeezed by the most information rather than starting at the top left, so first I redraw the grid with the bold regions marked (tool 1) so I can see which cells share a region. Each time a cell is forced, I write it in and it becomes new evidence, so the puzzle unlocks in a chain. At the end I read the finished table as a list of rows (tool 2) and look for a pattern (tool 5) as a final check.
1STEP 1
Redraw the grid and mark the five bold regions
Redraw the grid and mark the five regions.
5 × 5 = 25
The bold outlines are the part of the figure that is easiest to misread, so drawing them first turns a picture into a set of five explicit lists you can check against.
4.OA.C.5Draw A Diagram2STEP 2
Find the most squeezed cell: (1,4)
Start with the cell that has only one option.
{1,2,3,4,5} - {1,5,2} - {5,3,2} = {4}
A cell that meets a busy row and a busy column has almost no candidates left, so starting there costs no guessing at all.
4.OA.C.5Eliminate PossibilitiesStarting at the most squeezed cell is best, because it is where the fewest numbers are still allowed.
Why?
Each row, each column and each bold region holds every number exactly once, so a cell is barred from everything already placed near it.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
When those bars leave only one number standing, that number is forced and no guessing is needed.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
3STEP 3
Finish row 1, then use region B
Also use that each line adds to 15.
1+2+3+4+5 = 15, 15 - (1+5+4+2) = 3, 15 - (3+5+4+2) = 1
When four of the five numbers are showing, you do not have to test candidates at all -- subtracting from 15 names the missing one straight away.
2.OA.B.2Eliminate Possibilities4STEP 4
Finish column 4 and start column 1
Finish column 4 and start column 1.
15 - (4+5+3+2) = 1
Two blanks holding two known numbers is a fork with only two branches, and one clash in a row is enough to close one branch for good.
4.OA.C.5Eliminate Possibilities5STEP 5
Use region D to split 2 and 4
A region separates the 2 and the 4.
15 - (2+1+5+4) = 3
A region with two blanks behaves just like a row with two blanks: name the pair of missing numbers first, then let one outside clash decide which goes where.
2.OA.B.2Eliminate Possibilities6STEP 6
Fill column 3, then rows 3 and 4
Fill column 3, then rows 3 and 4.
15 - (5+3+2+4) = 1
Each newly written number makes its row, its column and its region tighter, so the deductions keep getting easier instead of harder.
4.OA.C.5Eliminate Possibilities7STEP 7
Close the last two cells
The last two cells fill themselves in.
15 - (5+4+2+1) = 3, 15 - (3+5+4+2) = 1
Because every single entry was forced, there was never a branch to come back to, so the finished table is unique.
2.OA.B.2Eliminate Possibilities8STEP 8
Read off the finished table and check the pattern
The finished table is each row shifted by one.
Spotting the hidden shift is a strong independent check: a slide-by-one table can never repeat a number in a column, so the whole column rule verifies itself in one look.
3.OA.D.9Look For A Pattern9STEP 9
Verify all fifteen conditions
All fifteen conditions check out.
Listing all fifteen groups is only fifteen quick reads, and it catches the one kind of mistake that logic chains are prone to -- a slip copied forward without noticing.
4.OA.C.5Make A Systematic ListAnswer
13542 / 21354 / 42135 / 54213 / 35421
The answer must be a table of the numbers 1 to 5, and it is: each of 1, 2, 3, 4, 5 appears exactly five times, so all 25 cells are used and 5 × 5 = 25 checks out. Every row, every column and every bold region sums to 1+2+3+4+5 = 15, and 15 × 5 = 75 is the total of the whole table whichever way you slice it -- by rows, by columns or by regions. All nine printed numbers are unchanged. No units are involved because these are labels in a puzzle, not measurements. And since every cell was forced rather than guessed, the table is not just a solution but the only one.
Takeaway
Do not start at the top left -- start at the cell that its row, its column and its bold region all argue about, because that cell usually has only one number left.
- Redraw the grid and mark the five bold regions
- Find the most squeezed cell: (1,4)
- Finish row 1, then use region B
- Finish column 4 and start column 1
- Use region D to split 2 and 4
- Fill column 3, then rows 3 and 4
- Close the last two cells
- Read off the finished table and check the pattern
- Verify all fifteen conditions