Math Puzzles Printable for Learning | Activity Shelter | Fun math ... - Free Printable

Explanation:
We are given four 3×3 grids with some numbers filled in. The instruction says to “use the 120 board” — that means we’re working with numbers from 1 to 120, and likely the grids follow a pattern like a hundreds chart (or 120 chart), where numbers increase by 1 as you go right, and by 10 as you go down (since it's base-10).

Let’s analyze each grid one by one.

---

Grid 1 (top-left):

```
? 46 ?
55 ? 57
? 66 ?
```

In a 120 chart:
- Moving right → +1
- Moving down → +10

Let’s find positions.

We know:
- Row 1, col 2 = 46
→ So row 1, col 1 = 46 − 1 = 45
→ Row 1, col 3 = 46 + 1 = 47

Row 2, col 1 = 55
Check: row 1, col 1 = 45 → down one row = 45 + 10 = 55 ✔ matches
So row 2, col 2 = 55 + 1 = 56
Row 2, col 3 = 57 (given) → 56 + 1 = 57 ✔

Row 3, col 2 = 66 (given)
Row 2, col 2 = 56 → down = 56 + 10 = 66 ✔
So row 3, col 1 = 66 − 1 = 65
Row 3, col 3 = 66 + 1 = 67

So Grid 1 completed:

```
45 46 47
55 56 57
65 66 67
```

---

Grid 2 (top-right):

```
? 99 ?
108 ? 110
? 119 ?
```

Row 1, col 2 = 99
→ Row 1, col 1 = 98
→ Row 1, col 3 = 100

Row 2, col 1 = 108
Check: 98 + 10 = 108 ✔
So row 2, col 2 = 108 + 1 = 109
Row 2, col 3 = 110 (given) → 109 + 1 = 110 ✔

Row 3, col 2 = 119
Row 2, col 2 = 109 → 109 + 10 = 119 ✔
So row 3, col 1 = 119 − 1 = 118
Row 3, col 3 = 119 + 1 = 120

Grid 2 completed:

```
98 99 100
108 109 110
118 119 120
```

---

Grid 3 (bottom-left):

```
91 ? ?
? ? 103
? 112 ?
```

Row 1, col 1 = 91
→ Row 1, col 2 = 92
→ Row 1, col 3 = 93

Row 3, col 2 = 112
So row 2, col 2 = 112 − 10 = 102
Then row 2, col 3 = 102 + 1 = 103 ✔ matches given.

So row 2, col 1 = 102 − 1 = 101

Now check row 1, col 3 = 93 → down to row 2, col 3 = 93 + 10 = 103 ✔
Then row 3, col 3 = 103 + 10 = 113

Row 3, col 1 = 101 + 10 = 111

Let’s fill all:

Row 1: 91, 92, 93
Row 2: 101, 102, 103
Row 3: 111, 112, 113

Check:
- 91 → down = 101 → down = 111 ✔
- 92 → down = 102 → down = 112 ✔
- 93 → down = 103 → down = 113 ✔

Grid 3 completed:

```
91 92 93
101 102 103
111 112 113
```

---

Grid 4 (bottom-right):

```
? 93 ?
? 104 ?
? ? 115
```

Row 1, col 2 = 93
→ Row 1, col 1 = 92
→ Row 1, col 3 = 94

Row 2, col 2 = 104
So row 1, col 2 = 93 → down = 93 + 10 = 103, but given is 104 — wait! That’s off by 1.

Hmm — maybe our assumption is wrong? Let’s double-check.

If row 2, col 2 = 104, then row 1, col 2 should be 104 − 10 = 94, but it’s given as 93. Contradiction.

Alternative: maybe the grid isn’t strictly aligned to rows/columns of 10? But the other grids fit perfectly with +10 down, +1 right.

Wait — maybe the grid is not starting at column 1 = same tens digit? Let’s try using known values to deduce positions.

We have:
- (1,2) = 93
- (2,2) = 104
- (3,3) = 115

From (1,2)=93 to (2,2)=104 → difference = +11
That suggests maybe the rows are offset? Unlikely.

Let’s instead assume the grid is still standard (down = +10, right = +1), and see if any value is misread.

Given:
Top row: blank, 93, blank
Middle: blank, 104, blank
Bottom: blank, blank, 115

Try to find what number must be above 104: 104 − 10 = 94 → so (1,2) should be 94, but it's written 93. Could it be a typo? But the worksheet is printed — more likely we need to use the pattern *within* the grid, not assume full 120 chart alignment.

Alternate approach: Use differences between known numbers.

We know:
- (2,2) = 104
- (3,3) = 115
So from (2,2) to (3,3): down 1, right 1 → should be +10 +1 = +11 → 104 + 11 = 115 ✔ matches!

So the rule holds: down = +10, right = +1.

Then (1,2) = 93
→ (2,2) should be 93 + 10 = 103, but it's 104. Conflict.

Unless… maybe (1,2) is not directly above (2,2)? But in a 3×3 grid, positions are aligned.

Wait — could the grid be numbered differently? Let's test using (3,3)=115.

If (3,3) = 115, then:
- (3,2) = 115 − 1 = 114
- (3,1) = 114 − 1 = 113
- (2,3) = 115 − 10 = 105
- (2,2) = 105 − 1 = 104 ✔ matches given
- (2,1) = 104 − 1 = 103
- (1,3) = 105 − 10 = 95
- (1,2) = 95 − 1 = 94
But given (1,2) = 93 — still off.

Unless the given 93 is actually 94? But it's clearly written 93.

Let me re-express all knowns precisely:

Grid 4:
Row 1: [ ?, 93, ? ]
Row 2: [ ?, 104, ? ]
Row 3: [ ?, ?, 115 ]

Assume standard chart: cell (r,c) = base + 10*(r−1) + (c−1)

Let unknown top-left = x
Then:
- (1,1) = x
- (1,2) = x + 1 = 93 → so x = 92
- (1,3) = x + 2 = 94

- (2,1) = x + 10 = 102
- (2,2) = x + 11 = 92 + 11 = 103, but given 104 → mismatch.

So either the grid is shifted, or one number is meant to be deduced differently.

Wait — look back at Grid 3: we had 91, ?, ? and 103, 112 — and it worked with +10 down.

What if in Grid 4, the middle number 104 is correct, and 93 is correct, so the vertical step is +11? But then (3,3) = 115, and (2,2)=104, difference = 11 → down+right = +11, which fits +10 down +1 right. So vertical step is still +10.

Then (1,2) must be 104 − 10 = 94. So likely the printed "93" is a mistake — but since this is a student worksheet, perhaps we’re supposed to ignore that and fill based on consistency with other clues.

Alternatively, maybe the grid is not aligned to columns — but that would be unusual.

Let’s try solving using only two solid anchors: (2,2)=104, (3,3)=115.

As above:
- (3,3) = 115
→ (3,2) = 114
→ (3,1) = 113
→ (2,3) = 105
→ (2,2) = 104 ✔
→ (2,1) = 103
→ (1,3) = 95
→ (1,2) = 94
→ (1,1) = 93

Ah! So (1,1) = 93, not (1,2). But the worksheet shows 93 in (1,2). Could it be misaligned in our reading?

Look again: In the image description, bottom-right grid is:

```
93
104
115
```

The layout in the problem says:
> 93
> 104
> 115

But positioned as:
- First row: blank, 93, blank
- Second row: blank, 104, blank
- Third row: blank, blank, 115

That matches our earlier.

But if we set (1,1) = 93, then (1,2)=94, (2,2)=104, (3,3)=115 — still (2,2) = 93 + 10 + 1 = 104 ✔! Wait:

If (1,1) = 93
→ (1,2) = 94
→ (2,2) = 94 + 10 = 104 ✔
→ (3,3) = 104 + 10 + 1 = 115 ✔

So the 93 should be in (1,1), but the worksheet places it in (1,2). Unless the student is expected to realize the pattern and correct based on consistency.

Given that all other grids follow strict +1 right, +10 down, and this one is the only inconsistent one, and the numbers 93, 104, 115 fit perfectly if 93 is top-left, I suspect a layout misinterpretation — but since the problem gives 93 in the middle of top row, we must honor that.

Let me try one more idea: maybe the grids are not all oriented the same — but no.

Wait — look at Grid 1: 46 is top-middle, and below it is 56 (not given, but we inferred), and 66 is bottom-middle — that’s +10 each time. So middle column increases by 10 per row.

In Grid 4: middle column has 93 (row1), 104 (row2), ? (row3)
93 → 104 is +11, so row3 middle should be 104 + 11 = 115 — but 115 is bottom-right, not bottom-middle.

But bottom-right is 115. If bottom-middle were 115, then bottom-right = 116.

Given bottom-right is 115, and middle of row2 is 104, then middle of row3 = 115 − 1 = 114 (if same row), but it's not same row.

Let’s just solve by completing all cells that can be deduced uniquely from two knowns in same row/column.

From Grid 4:
- We know (2,2) = 104
- (3,3) = 115
→ So (3,2) = 115 − 1 = 114
→ (2,3) = 115 − 10 = 105
→ (2,2) = 105 − 1 = 104 ✔
→ (1,3) = 105 − 10 = 95
→ (1,2) = 95 − 1 = 94
But worksheet says 93. Close — maybe it's a typo, and it should be 94.

Given that in educational worksheets, such small errors happen, and all other grids are consistent with +1/+10, the intended answer likely assumes standard chart rules.

Moreover, in Grid 3: 91, then 103 is two rows down and two columns right: 91 + 20 + 2 = 113, but 103 is given in row2 col3, and 91 is row1 col1: 91 +10 +2 = 103 ✔.

So rule holds.

Thus for Grid 4, to satisfy (2,2)=104 and (3,3)=115, we must have:
- (1,2) = 94
But it's printed 93 — possibly a distractor, or the student is to notice and use the 120 board to verify.

Since the instruction says “Use the 120 board to help you fill in the charts”, the student should look at a 120 chart and locate the numbers.

Let me quickly simulate a 120 chart snippet:

Rows of 10:
Row 9: 91 92 93 94 95 96 97 98 99 100
Row 10: 101 102 103 104 105 106 107 108 109 110
Row 11: 111 112 113 114 115 116 117 118 119 120

Now locate:
- 93 is at row9, col3
- 104 is at row10, col4
- 115 is at row11, col5

So they are diagonal: (9,3), (10,4), (11,5) — i.e., each step +1 row, +1 col → +11 each time: 93 → 104 → 115 ✔

So the grid is not aligned to columns 1-3 of the chart, but rather a diagonal slice.

In the 3×3 grid, the positions correspond to:
- (1,1) = row9, col3 = 93
- (1,2) = row9, col4 = 94
- (1,3) = row9, col5 = 95
- (2,1) = row10, col3 = 103
- (2,2) = row10, col4 = 104
- (2,3) = row10, col5 = 105
- (3,1) = row11, col3 = 113
- (3,2) = row11, col4 = 114
- (3,3) = row11, col5 = 115

But the worksheet shows 93 in (1,2), not (1,1). However, looking at the visual layout in the original image (which we can’t see, but based on description), it's possible the 93 is actually in top-left of that grid — the text may have been misaligned in transcription.

Given that 93, 104, 115 form a perfect +11 diagonal, and the only consistent completion is:

```
93 94 95
103 104 105
113 114 115
```

And this matches the 120 chart perfectly.

Also, in Grid 1: 46,55,57,66 — let's verify in chart:
Row4: 41-50 → 46 is col6
Row5: 51-60 → 55 is col5, 57 col7
Row6: 61-70 → 66 col6
Not a straight column, but our earlier fill worked.

Given time, and that three grids are fully consistent with +1/+10, and the fourth becomes consistent if we place:
- top row: 93, 94, 95
- middle: 103, 104, 105
- bottom: 113, 114, 115

And the given numbers are 93 (should be top-left), 104 (center), 115 (bottom-right) — which matches this.

I think the worksheet intends that, and the "93" is in the first cell of that grid (top-left), even if formatting suggests middle.

Since the student is to use the 120 board, they would look up and see that 93, then next row same column is 103, etc.

Thus, final fills:

Grid 1:
45 46 47
55 56 57
65 66 67

Grid 2:
98 99 100
108 109 110
118 119 120

Grid 3:
91 92 93
101 102 103
111 112 113

Grid 4:
93 94 95
103 104 105
113 114 115

All numbers are within 1–120, and patterns hold.

Final Answer:
Grid 1: 45, 46, 47 / 55, 56, 57 / 65, 66, 67
Grid 2: 98, 99, 100 / 108, 109, 110 / 118, 119, 120
Grid 3: 91, 92, 93 / 101, 102, 103 / 111, 112, 113
Grid 4: 93, 94, 95 / 103, 104, 105 / 113, 114, 115