Evaluations for input

So I've got this equation: 1 + 2 * 3 + 4 * 5 + 6 * 7 + 8 * 9 = 479, and it's supposed to be incorrect, but I need to add just one pair of parentheses to make it true. Hmm, let's see.

First, I should probably understand what the equation is currently doing. In math, multiplication is performed before addition according to the order of operations (PEMDAS/BODMAS rules). So, in the expression 1 + 2 * 3 + 4 * 5 + 6 * 7 + 8 * 9, the multiplications would be calculated first, and then the additions.

Let's compute it step by step without any parentheses:

First, the multiplications:

2 * 3 = 6

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

Now, plug these back into the equation:

1 + 6 + 20 + 42 + 72

Now, add them up:

1 + 6 = 7

7 + 20 = 27

27 + 42 = 69

69 + 72 = 141

So, without any parentheses, the equation equals 141, not 479. So, clearly, something needs to change.

Now, I need to add one pair of parentheses to make this equal to 479. Parentheses can change the order of operations, so whatever is inside the parentheses gets calculated first.

I need to think about where to place these parentheses to alter the result significantly, since 141 is quite a bit smaller than 479.

Maybe I can group some additions and multiplications in a way that increases the result.

Let me consider different placements:

Option 1: (1 + 2) * 3 + 4 * 5 + 6 * 7 + 8 * 9

Calculate inside the parentheses first:

1 + 2 = 3

Then, 3 * 3 = 9

Now, the equation is: 9 + 4 * 5 + 6 * 7 + 8 * 9

Next, multiplications:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 9 + 20 + 42 + 72

Add them up:

9 + 20 = 29

29 + 42 = 71

71 + 72 = 143

Still not 479.

Option 2: 1 + (2 * 3 + 4) * 5 + 6 * 7 + 8 * 9

First, inside the parentheses:

2 * 3 = 6

6 + 4 = 10

Now, multiply by 5: 10 * 5 = 50

So, the equation becomes: 1 + 50 + 6 * 7 + 8 * 9

Next, multiplications:

6 * 7 = 42

8 * 9 = 72

Now, add them up:

1 + 50 = 51

51 + 42 = 93

93 + 72 = 165

Still not 479.

Option 3: 1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9

First, inside the parentheses:

3 + 4 = 7

Then, 2 * 7 = 14

Then, 14 * 5 = 70

So, the equation is: 1 + 70 + 6 * 7 + 8 * 9

Next, multiplications:

6 * 7 = 42

8 * 9 = 72

Now, add them up:

1 + 70 = 71

71 + 42 = 113

113 + 72 = 185

Still not 479.

Option 4: 1 + 2 * 3 + 4 * (5 + 6) * 7 + 8 * 9

First, inside the parentheses:

5 + 6 = 11

Then, 11 * 7 = 77

So, the equation is: 1 + 2 * 3 + 4 * 77 + 8 * 9

Next, multiplications:

2 * 3 = 6

4 * 77 = 308

8 * 9 = 72

Now, add them up:

1 + 6 = 7

7 + 308 = 315

315 + 72 = 387

Closer, but still not 479.

Option 5: 1 + 2 * 3 + 4 * 5 + 6 * (7 + 8) * 9

First, inside the parentheses:

7 + 8 = 15

Then, 15 * 9 = 135

So, the equation is: 1 + 2 * 3 + 4 * 5 + 6 * 135

Next, multiplications:

2 * 3 = 6

4 * 5 = 20

6 * 135 = 810

Now, add them up:

1 + 6 = 7

7 + 20 = 27

27 + 810 = 837

Way too big.

Hmm, 837 is much larger than 479. Maybe I need to reduce the impact of the parentheses.

Option 6: 1 + 2 * (3 + 4 * 5) + 6 * 7 + 8 * 9

First, inside the parentheses:

3 + 4 * 5

Wait, in this case, multiplication comes before addition inside the parentheses, so 4 * 5 = 20, then 3 + 20 = 23

Then, 2 * 23 = 46

So, the equation is: 1 + 46 + 6 * 7 + 8 * 9

Next, multiplications:

6 * 7 = 42

8 * 9 = 72

Now, add them up:

1 + 46 = 47

47 + 42 = 89

89 + 72 = 161

Still not 479.

Option 7: 1 + 2 * 3 + 4 * (5 + 6 * 7) + 8 * 9

First, inside the parentheses:

5 + 6 * 7

Multiplication first: 6 * 7 = 42, then 5 + 42 = 47

Then, 4 * 47 = 188

So, the equation is: 1 + 2 * 3 + 188 + 8 * 9

Next, multiplications:

2 * 3 = 6

8 * 9 = 72

Now, add them up:

1 + 6 = 7

7 + 188 = 195

195 + 72 = 267

Still not 479.

Option 8: 1 + 2 * 3 + 4 * 5 + 6 * (7 + 8 * 9)

First, inside the parentheses:

7 + 8 * 9

Multiplication first: 8 * 9 = 72, then 7 + 72 = 79

Then, 6 * 79 = 474

So, the equation is: 1 + 2 * 3 + 4 * 5 + 474

Next, multiplications:

2 * 3 = 6

4 * 5 = 20

Now, add them up:

1 + 6 = 7

7 + 20 = 27

27 + 474 = 479

Perfect! That equals 479.

So, by placing the parentheses around 7 + 8 * 9, and multiplying that sum by 6, we get the desired result.

Therefore, the corrected equation is: 1 + 2 * 3 + 4 * 5 + 6 * (7 + 8 * 9) = 479

Final Answer

[ \boxed{1 + 2 \times 3 + 4 \times 5 + 6 \times (7 + 8 \times 9) = 479} ]