Tenner Grids addition puzzles for primary and secondary math learners - Free Printable

Let’s look at the grid:

We have a 3x3 grid with some numbers filled in:

Row 1: 5, __, 8
Row 2: 0, __, 1
Row 3: 5, 8, 9

The bottom row is shaded — maybe that’s a hint. Let’s see if there’s a pattern going down each column.

Check Column 1:
Top: 5
Middle: 0
Bottom: 5
→ 5 + 0 = 5? That works!

Check Column 3:
Top: 8
Middle: 1
Bottom: 9
→ 8 + 1 = 9? Yes! That also works.

So it looks like:
Top number + Middle number = Bottom number (for each column)

Now let’s apply that to Column 2.

Column 2:
Top: ?
Middle: ?
Bottom: 8

Wait — we don’t know top or middle yet. But maybe we can find them using another pattern?

Hold on — maybe the rows have patterns too?

Look at Row 1: 5, ?, 8 → difference between 5 and 8 is 3. Maybe middle is 6 or 7? Not sure.

Row 2: 0, ?, 1 → difference is 1. Maybe middle is 0.5? Doesn’t seem right for whole numbers.

But earlier, the column rule worked perfectly for columns 1 and 3.

Let’s assume the same rule applies to column 2:
Top + Middle = Bottom → Top + Middle = 8

But we still have two unknowns. Is there another clue?

Wait — look at the entire grid again.

Maybe the sum of each row is the same?

Row 3: 5 + 8 + 9 = 22

If all rows sum to 22...

Row 1: 5 + ? + 8 = 22 → ? = 22 - 13 = 9

Row 2: 0 + ? + 1 = 22 → ? = 21

That gives us:

Row 1: 5, 9, 8
Row 2: 0, 21, 1
Row 3: 5, 8, 9

Now check columns with this:

Column 1: 5 + 0 = 5 ✔️
Column 2: 9 + 21 = 30 ≠ 8 ✘ → doesn’t work.

So row sums being equal doesn’t fit the column rule.

Back to column rule: Top + Middle = Bottom

For column 2: Top + Middle = 8

Is there any other constraint?

What if we look diagonals? Or maybe the center cell is special?

Another idea: Maybe the grid is built so that each column follows Top + Middle = Bottom, and we’re missing only the center cell? But no — both center-top and center-middle are blank.

Wait — perhaps I misread. Let me re-express the grid clearly:

Let’s label positions:

A B C
D E F
G H I

Given:

A=5, C=8
D=0, F=1
G=5, H=8, I=9

We need to find B and E.

From column 1: A + D = G → 5 + 0 = 5 ✔️
Column 3: C + F = I → 8 + 1 = 9 ✔️
So column 2: B + E = H → B + E = 8

Still two variables.

But maybe there’s a row pattern?

Row 1: A + B + C = 5 + B + 8 = 13 + B
Row 2: D + E + F = 0 + E + 1 = 1 + E
Row 3: 5 + 8 + 9 = 22

If rows must be equal: 13 + B = 1 + E = 22

Then:

13 + B = 22 → B = 9
1 + E = 22 → E = 21

Then check column 2: B + E = 9 + 21 = 30 ≠ 8 → contradiction.

So rows aren’t equal.

Alternative idea: Maybe the center cell E is the average or something?

Or perhaps the grid has a magic square property? Unlikely since numbers repeat.

Wait — what if we consider that the bottom row is given, and the rule is strictly per column: Top + Middle = Bottom.

Then for column 2, we have B + E = 8.

But we need another equation.

Unless... is there a typo? Or maybe I’m overcomplicating.

Let me try assuming that the middle row is related to the top and bottom.

Another thought: Look at differences vertically.

Column 1: from top to bottom: 5 → 0 → 5 → change of -5, then +5
Column 3: 8 → 1 → 9 → change of -7, then +8 — not consistent.

Perhaps horizontal patterns within rows?

Row 1: 5 to 8 is +3, so maybe middle is 6.5? No.

Wait — here’s a new idea: What if the number in the center of the grid (E) is such that it completes both row and column logic?

But we still lack info.

Let me list what we know:

We must satisfy:
- Col1: 5 + 0 = 5 ✔️
- Col3: 8 + 1 = 9 ✔️
- Col2: B + E = 8

Also, perhaps the sum of all numbers? Total sum of known numbers: 5+8+0+1+5+8+9 = 36, plus B and E.

Total cells: 9, sum would be 36 + B + E.

But without target sum, not helpful.

Wait — what if the puzzle is designed so that the missing numbers are single digits, and B + E = 8, and perhaps they are symmetric or something?

Possible pairs for (B,E): (0,8), (1,7), (2,6), (3,5), (4,4), (5,3), etc.

Now look at existing numbers: 5,8,0,1,5,8,9 — so 0,1,5,8,9 are used.

Avoid duplicates? Not necessarily.

But notice: in row 1, we have 5 and 8 — if B=3, then 5,3,8 — nice sequence? 5 to 3 is -2, 3 to 8 is +5 — not great.

If B=4, then 5,4,8 — no obvious pattern.

Another approach: Perhaps the center cell E is the key, and it's the same as the bottom middle? H=8, but E is above it.

Wait — let's think differently.

What if the operation is not addition, but something else?

Col1: 5 and 0 make 5 — could be max(5,0)=5, min=0, but 5 is bottom.

Col3: 8 and 1 make 9 — 8+1=9, which worked.

Col2: B and E make 8.

But also, look at the values: in col1, top and bottom are both 5, middle is 0.

In col3, top 8, bottom 9, middle 1.

Notice that in col1: bottom = top, and middle = 0.

In col3: bottom = top +1, middle =1.

Not consistent.

Another idea: Perhaps the middle row is the difference between bottom and top?

Col1: bottom - top = 5 - 5 = 0, which matches middle D=0.

Col3: bottom - top = 9 - 8 = 1, which matches F=1.

Oh! That's it!

So for each column: Middle = Bottom - Top

Let's verify:

Col1: Bottom G=5, Top A=5, Middle D=5-5=0 ✔️

Col3: Bottom I=9, Top C=8, Middle F=9-8=1 ✔️

Perfect!

So for Col2: Bottom H=8, Top B=?, Middle E=?

According to the rule: Middle = Bottom - Top → E = 8 - B

But we also have from earlier assumption? No, now we have a clear rule.

Actually, the rule is: For each column, the middle cell equals the bottom cell minus the top cell.

So for column 2: E = H - B = 8 - B

But we still have two variables. However, in the grid, B and E are both unknown, but perhaps there's only one pair that makes sense with the rest of the grid? Or maybe we need to use row constraints.

Wait — do we have any other information? The problem is to fill in the blanks, implying there is a unique solution.

Perhaps the rows have a pattern too.

Let's assume the rule is only column-based: Middle = Bottom - Top

Then for col2: E = 8 - B

But we need another equation.

Unless... is there a possibility that the top row or middle row has a fixed sum or something?

Let's calculate what we have.

Suppose we let B be x, then E = 8 - x

Now look at row 1: 5, x, 8

Row 2: 0, 8-x, 1

Row 3: 5,8,9

Is there a relationship between rows?

For example, row 1 + row 2 = row 3? Let's see:

Col1: 5 + 0 = 5 ✔️

Col2: x + (8 - x) = 8 ✔️

Col3: 8 + 1 = 9 ✔️

Oh my goodness! It works for any x!

Because x + (8 - x) = 8, always.

So the column rule "Middle = Bottom - Top" implies that "Top + Middle = Bottom", which we had earlier, and also means that for each column, Top + Middle = Bottom, so when you add row1 and row2, you get row3.

And indeed:

Row1 + Row2 = [5+0, x+(8-x), 8+1] = [5,8,9] = Row3

So it holds for ANY value of x! But that can't be — the puzzle must have a unique answer.

Unless I missed something.

Look back at the image description: the bottom row is shaded. Maybe that indicates it's the result, and we need to find the inputs.

But still, mathematically, any B and E such that B + E = 8 will work.

But that doesn't make sense for a homework problem; there must be a specific answer.

Perhaps there's a diagonal pattern or something else.

Let me try to see if the center cell E is determined by other means.

Another idea: Perhaps the grid is a magic square where all rows, columns, and diagonals sum to the same number.

Row3 sum: 5+8+9=22

If magic square, all rows sum to 22.

Then row1: 5 + B + 8 = 22 → B=9

Row2: 0 + E + 1 = 22 → E=21

Then check columns:

Col1: 5+0+5=10 ≠22 — not magic square.

Diagonals: main diagonal A+E+I =5 + E +9=14+E

Other diagonal C+E+G=8+E+5=13+E

Set equal: 14+E =13+E → 14=13, impossible.

So not magic square.

Perhaps only the columns follow the rule, and the rows don't matter, but then why are there two blanks?

Unless the problem is to realize that B and E can be any numbers summing to 8, but that seems unlikely.

Wait — let's look at the numbers already present.

In the grid, the numbers are: 5,8,0,1,5,8,9

Notice that 5 and 8 appear twice.

Perhaps the missing numbers are to be chosen from unused digits or something.

Unused digits: 2,3,4,6,7

B + E =8, possible pairs from unused: 2+6, 3+5 but 5 used, 4+4, 6+2, 7+1 but 1 used.

So possible: (2,6), (6,2), (4,4)

Now, which one fits best?

Let's try B=2, E=6

Grid:

5 2 8

0 6 1

5 8 9

Check if there's a row pattern.

Row1: 5,2,8 — differences: -3, +6

Row2: 0,6,1 — +6, -5

Not nice.

B=6, E=2:

5 6 8

0 2 1

5 8 9

Row1: 5 to 6 is +1, 6 to 8 is +2

Row2: 0 to 2 is +2, 2 to 1 is -1

Not consistent.

B=4, E=4:

5 4 8

0 4 1

5 8 9

Row1: 5,4,8 — -1, +4

Row2: 0,4,1 — +4, -3

Still not great.

Perhaps the center cell should be the average or median.

Another thought: In many such puzzles, the center cell is often the key, and might be related to the surrounding cells.

For example, in some grids, the center is the sum of opposites or something.

Let's calculate the sum of all corner cells: A+C+G+I =5+8+5+9=27

Sum of edge centers: B,D,F,H = B+0+1+8 = B+9

Center E.

Total sum = 27 + (B+9) + E = 36 + B + E

But B+E=8, so total sum = 44

Not helpful.

Perhaps the product or other operations.

Let's go back to the initial insight.

We have for each column: Top + Middle = Bottom

This is satisfied as long as B + E = 8.

But perhaps in the context of the problem, since the bottom row is given and shaded, and the top and middle are to be found, but there's no additional constraint, maybe the intended answer is to recognize that the middle row is the difference, and for column 2, since bottom is 8, and if we assume top is the same as in other columns or something.

Notice that in column 1, top and bottom are both 5, middle is 0.

In column 3, top 8, bottom 9, middle 1.

In column 2, bottom is 8, so if we want symmetry, perhaps top is 8, then middle is 0, but 8+0=8, good, but then B=8, E=0.

Let's try that.

B=8, E=0

Grid:

5 8 8

0 0 1

5 8 9

Now check rows:

Row1: 5,8,8 — sum 21

Row2: 0,0,1 — sum 1

Row3: 22

Not equal.

Columns: col2: 8+0=8 ✔️

But now we have duplicate 8's, and 0's.

Is there a reason to choose this?

Perhaps the top row should be increasing or something.

Another idea: Perhaps the number in the center E is the same as the number below it or above it.

H=8, so if E=8, then B=0, since B+E=8.

Try B=0, E=8

Grid:

5 0 8

0 8 1

5 8 9

Now this looks interesting.

Row1: 5,0,8

Row2: 0,8,1

Row3: 5,8,9

Notice that row2 is almost the reverse of row1 or something.

Also, look at the columns:

Col1: 5,0,5

Col2: 0,8,8

Col3: 8,1,9

All satisfy Top + Middle = Bottom: 5+0=5, 0+8=8, 8+1=9 ✔️

Moreover, look at the grid:

5 0 8

0 8 1

5 8 9

Is there a pattern in the rows? Row1 and row2 share the 0 and 8 in different positions.

Also, the center cell E=8, which is the same as H=8, and C=8, so 8 appears three times, which is fine.

Similarly, 5 appears in A,G, and 0 in D,B, 1 in F, 9 in I.

Numbers used: 0,1,5,8,9 — all reasonable.

Perhaps this is the intended solution.

Why did I think of B=0, E=8? Because if I set E=8 (same as bottom), then B=0.

Or because in col2, if middle is 8, same as bottom, then top must be 0.

And 0 is already used in col1 middle, so it's consistent.

Moreover, in this case, the grid has a kind of symmetry.

Let me see if there's another common choice.

Earlier with B=9, E= -1, not good.

Or B=1, E=7, etc.

But B=0, E=8 seems plausible.

Perhaps the problem expects us to use the column rule and leave it, but since it's a fill-in-the-blank, likely specific numbers.

Another way: perhaps the sum of the top row equals sum of middle row or something.

With B=0, E=8:

Row1 sum: 5+0+8=13

Row2 sum: 0+8+1=9

Not equal.

Difference is 4.

Not helpful.

Perhaps the product.

I recall that in some puzzles, the center cell is the average of the four corners or something.

Corners: A=5, C=8, G=5, I=9, sum=27, average=6.75, not integer.

Median of corners: 5,5,8,9 -> median 6.5.

Not helping.

Let's consider the possibility that the missing numbers are to be found such that the grid has no repeated numbers, but 5 and 8 are already repeated, so probably not.

In the grid with B=0, E=8, we have:

Positions:
A=5, B=0, C=8
D=0, E=8, F=1
G=5, H=8, I=9

So numbers: 0 appears at B and D, 5 at A and G, 8 at C,E,H — so 8 appears three times, 0 and 5 twice, 1 and 9 once.

Seems acceptable.

Perhaps the intended answer is B=3, E=5, but 5 is already used, and 3+5=8.

Grid:

5 3 8

0 5 1

5 8 9

Then col2: 3+5=8 ✔️

But now 5 appears at A,G,E — three times, same as before.

No advantage.

I think the most symmetric or logical choice is B=0, E=8, because then the middle row has 0,8,1, and top row has 5,0,8, so they share 0 and 8, and bottom has 5,8,9.

Also, in col2, top is 0, middle is 8, bottom is 8, which is consistent with the rule.

Moreover, if we look at the first column: top 5, middle 0, bottom 5 — so middle is 0, which is the difference.

Second column: top 0, middle 8, bottom 8 — difference is 8, which is middle.

Third column: top 8, middle 1, bottom 9 — difference 1, which is middle.

So the rule "middle = bottom - top" is satisfied, and for col2, if top is 0, then middle = 8 - 0 = 8, which is consistent.

And 0 is a nice number, already used in the grid.

Perhaps the problem is from a context where 0 is expected.

I recall that in some similar puzzles, the missing number is 0.

So I'll go with B=0, E=8.

Therefore, the completed grid is:

5 0 8

0 8 1

5 8 9

So the missing numbers are 0 and 8.

But the question is to solve the problem, and the problem is to fill in the blanks, so probably report the values.

Since the user didn't specify what to output, but in the context, likely the values for the empty cells.

In the image, there are two empty cells: top-middle and center.

So answers are 0 and 8.

To confirm, let's see if there's a standard solution.

Upon second thought, in the column rule "Top + Middle = Bottom", for col2, if we assume that the top cell is the same as the middle cell of col1 or something, but col1 middle is 0, so B=0, then E=8.

Yes.

Perhaps the center cell is the sum of the adjacent cells or something, but let's not overcomplicate.

I think B=0, E=8 is the intended answer.

So final answer: the top-middle cell is 0, and the center cell is 8.

But how to present it? Probably as the values.

Since the problem might expect the grid filled, but in text, we can say the missing numbers are 0 and 8.

Looking back at the user's request: "Solve the problem accurately." and "provide the final correct answer".

Probably, since it's a grid with two blanks, the answer is the two numbers.

Perhaps in order: first blank (top-middle), second blank (center).

So 0 and 8.

To be precise, let's box the answer.

Final Answer: 0 and 8

But let's write it clearly.