4th Grade Math Puzzles - Free Printable

Actually, there is a mistake in the problem statement for Part 1.

Let’s carefully analyze both parts.

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## 🔍 PART 1: Magic Square with numbers -4, -3, -2, -1, 1, 2, 3, 4 and center = 0

You are told to place these 8 numbers + 0 (which is already placed in the center) into a 3x3 grid so that each row, column, and diagonal adds to 15.

But let’s check the total sum of all numbers:

Numbers: -4, -3, -2, -1, 0, 1, 2, 3, 4
Sum = (-4 + 4) + (-3 + 3) + (-2 + 2) + (-1 + 1) + 0 = 0

In a 3x3 magic square, if each row sums to S, then total sum of all 9 cells = 3 × S.

So if each row sums to 15 → total sum = 45.
But our numbers only sum to 0 — this is impossible.

✔ So it’s impossible to make a magic square with these numbers adding to 15.

But wait — the worksheet says “Total must be 0” — which is correct — and also says “adds up to 15”. That’s a contradiction.

➡️ This is likely a typo. The intended target sum should be 0, not 15.

Because:
- Total sum = 0
- 3 rows → each row should sum to 0 ÷ 3 = 0

This makes sense!

Also, in standard 3x3 magic squares using numbers symmetric around 0 (like -4 to 4), the magic constant is always 0.

---

## ✔ Corrected Problem 1: Make a magic square using -4, -3, -2, -1, 0, 1, 2, 3, 4 so that each row, column, diagonal sums to 0.

We can build it by modifying the classic 1–9 magic square.

Classic 3x3 magic square (sum = 15):

```
8 1 6
3 5 7
4 9 2
```

To convert it to use numbers from -4 to 4, subtract 5 from every number:

8-5=3, 1-5=-4, 6-5=1
3-5=-2, 5-5=0, 7-5=2
4-5=-1, 9-5=4, 2-5=-3

New square:

```
3 -4 1
-2 0 2
-1 4 -3
```

✔ Check row sums:
- Row 1: 3 + (-4) + 1 = 0
- Row 2: -2 + 0 + 2 = 0
- Row 3: -1 + 4 + (-3) = 0

✔ Columns:
- Col 1: 3 + (-2) + (-1) = 0
- Col 2: -4 + 0 + 4 = 0
- Col 3: 1 + 2 + (-3) = 0

✔ Diagonals:
- 3 + 0 + (-3) = 0
- 1 + 0 + (-1) = 0

Perfect!

And center is 0 — as given.

---

## 🧩 ANSWER TO PART 1:

Fill the grid as follows:

```
[ 3 ] [ -4 ] [ 1 ]
[ -2] [ 0 ] [ 2 ]
[ -1] [ 4 ] [ -3]
```

---

## 🔍 PART 2: Use digits -6, -5, -4, -3, -2, -1, 1, 2 and center = 0 to make each line sum to -6.

Wait — we have 8 numbers + 0 = 9 numbers.

List: -6, -5, -4, -3, -2, -1, 0, 1, 2

Sum = (-6) + (-5) + (-4) + (-3) + (-2) + (-1) + 0 + 1 + 2
= Let's group: (-6 -5 -4 -3 -2 -1) + (0 + 1 + 2)
= (-21) + (3) = -18

If each row sums to -6, total = 3 × (-6) = -18 → ✔ matches!

So this is possible.

We need to arrange them so each row, column, diagonal = -6.

Again, we can adapt the classic magic square.

Classic 1–9 magic square (sum=15):

```
8 1 6
3 5 7
4 9 2
```

We want numbers summing to -18 total, and magic constant -6.

Average per cell = -18 ÷ 9 = -2.

So we can shift the classic square down by 7? Let’s find the mapping.

In classic square, numbers are 1 to 9. We want to map them to our set: -6, -5, -4, -3, -2, -1, 0, 1, 2.

Note: Our numbers are from -6 to 2 — that’s 9 numbers: -6, -5, -4, -3, -2, -1, 0, 1, 2.

Compare to 1–9: subtract 7 from each number in classic square:

8-7=1, 1-7=-6, 6-7=-1
3-7=-4, 5-7=-2, 7-7=0
4-7=-3, 9-7=2, 2-7=-5

So new square:

```
1 -6 -1
-4 -2 0
-3 2 -5
```

✔ Check row sums:
- Row 1: 1 + (-6) + (-1) = -6
- Row 2: -4 + (-2) + 0 = -6
- Row 3: -3 + 2 + (-5) = -6

✔ Columns:
- Col 1: 1 + (-4) + (-3) = -6
- Col 2: -6 + (-2) + 2 = -6
- Col 3: -1 + 0 + (-5) = -6

✔ Diagonals:
- 1 + (-2) + (-5) = -6
- -1 + (-2) + (-3) = -6

Center is -2 — but the problem says center must be 0.

Uh oh! In our square, center is -2, but problem says “0” is already placed in center.

So we need to adjust.

Let’s try another approach.

We know center must be 0.

In a 3x3 magic square, the center is always the average of all numbers.

Our numbers: -6, -5, -4, -3, -2, -1, 0, 1, 2 → average = -18/9 = -2

But the problem forces center = 0 — which is not the average.

That means: It’s impossible to make a magic square with these numbers and center = 0, because in any 3x3 magic square, the center must equal the average of all numbers.

Why?

Because in a magic square, every row, column, diagonal sums to S.

The center is part of 1 row, 1 column, and 2 diagonals — total 4 lines.

But more simply: the sum of all rows = 3S.

Also, if you add the middle row, middle column, and two diagonals, you get 4S, but the center is counted 4 times, and other cells once or twice.

Standard proof: The center value = S/3.

Since total sum = 3S, and average = S/3 → center = average.

Here, average = -2 → center must be -2.

But problem says center = 0 → contradiction.

✔ Therefore, Part 2 is impossible as stated.

---

## ❗ BUT WAIT — maybe the problem meant to say "Total must be -6" meaning the magic constant is -6, and center is 0 — but that contradicts math.

Alternatively, perhaps the list of numbers is wrong.

Let’s double-check the problem:

> "using the digits -6, -5, -4, -3, -2, -1, 1, 2 so that each line adds up to -6?"

Wait — it lists 8 numbers: -6, -5, -4, -3, -2, -1, 1, 2 — and says center is 0 — so total 9 numbers.

Sum = -6-5-4-3-2-1+1+2+0 = let’s compute:

Negative: -6-5-4-3-2-1 = -21
Positive: 1+2 = 3
Plus 0 → total = -21 + 3 = -18

Magic constant = -6 → total = 3×(-6) = -18 → ✔ matches.

But center must be average = -2 — not 0.

So unless we ignore the center constraint, it’s impossible.

Perhaps the problem has a typo — maybe center should be -2?

Or maybe the target sum is different?

Alternatively, maybe they meant to include -7 or exclude something?

But as written — with center = 0 — it’s impossible.

---

## 🎯 FINAL ANSWERS:

✔ PART 1 (corrected to sum to 0):

```
[ 3 ] [ -4 ] [ 1 ]
[ -2] [ 0 ] [ 2 ]
[ -1] [ 4 ] [ -3]
```

*(Note: The worksheet incorrectly says “adds up to 15” — it should be 0.)*

---

✘ PART 2: Impossible as stated.

Because:
- Numbers sum to -18 → magic constant must be -6 → OK.
- But center must be average = -2, not 0.
- Since center is forced to be 0, no solution exists.

---

## 💡 TEACHER NOTE:

This worksheet contains an error in Part 1 (target sum 15 instead of 0) and Part 2 (center=0 contradicts math). For educational purposes, you might want to correct Part 1 to target sum 0, and either remove the center constraint in Part 2 or change the center to -2.

---

## ✔ If we ignore the center constraint in Part 2 and just make a magic square with those numbers summing to -6, here’s one:

Using the earlier mapping:

```
[ 1 ] [ -6 ] [ -1 ]
[ -4] [ -2 ] [ 0 ]
[ -3] [ 2 ] [ -5 ]
```

Sum per row/column/diagonal = -6.

Center is -2 — not 0.

If you must have center=0, then no solution exists.

---

## 📝 SUMMARY:

Part 1 Solution (with corrected target sum 0):

```
3 -4 1
-2 0 2
-1 4 -3
```

Part 2: No solution possible with center = 0.

---

Let me know if you’d like to see how to fix the worksheet or create a valid version!