# Blog

## Secret pairs

By

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Difficulty: Extreme!

Lavonne is testing a secret code with her friend Jack. First, she picks 3 different secret numbers between 1 and 9. Then she finds the sum of each pair of numbers. For example, secret numbers 2, 3, and 6 give these three sums: 5 (2+3), 8 (2+6), and 9 (3+6).

Lavonne wants to know if Jack can figure out the original secret numbers just from the sums.

If she tells him that the sums are 7, 12, and 13, can Jack recreate the secret numbers?

## Need a hint?

All 3 secret numbers are different. So one is the smallest, one is the largest, and one is in between.

7 is the smallest total, so it must be the 2 smallest secret numbers added together.

13 is the largest total, so it must be the 2 largest secret numbers added together.

12 is the remaining total, so it must be the largest and smallest secret numbers added together.

Remember that the secret numbers have to be between 1 and 9. This may help you eliminate some options.

Lavonne’s secret numbers are 3, 4 and 9.

To solve this problem, it can be helpful to look at the example first, which helpfully arranges all the numbers from smallest to largest. Secret numbers 2, 3, and 6 give these three sums:

5 (2+3), which is the smallest secret number plus the middle secret number.

8 (2+6), which is the smallest secret number plus the largest secret number.

9 (3+6), which is the middle secret number plus the largest secret number.

Notice that the smallest and largest sums both involve the middle secret number. As you’ll see in the next few steps, we can use this observation to narrow down the options.

We can turn our attention to the new set of secret numbers by looking at their sums, 7, 12, and 13. Let’s begin by brainstorming possible number pairs that add up to 7:

7 = 1+6, here 1 is the smallest secret number and 6 is the middle secret number

7 = 2+5, here 2 is the smallest secret number and 5 is the middle secret number

7 = 3+4, here 3 is the smallest secret number and 4 is the middle secret number

To help us narrow down the options, we can use the possible middle secret numbers above to brainstorm possible pairs for the largest sum, 13. As we saw in the example above, the middle sum 12 must be the sum of the smallest secret number plus the middle secret number.

13 = 6+7, with 6 as the middle secret number, the largest secret number must be 7. As we found above, when the middle secret number is 6, the smallest secret number must be 1. But 1 and 6 don’t add up to 12.

13 = 5+8, with 5 as the middle secret number, the largest secret number must be 8. When the middle secret number is 5, the smallest secret number must be 2. But 2 and 8 don’t add up to 12.

13 = 4+9, with 4 as the middle secret number, the largest secret number must be 9. When the middle secret number is 4, the smallest secret number must be 3. Finally, 3 and 9 do add up to 12!

This makes 3, 4, and 9 the secret numbers and we eliminated all other possibilities.

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## 4 responses

1. Martin

If we use letters instead of numbers, we can say the secret numbers are A, B, and C.

The three sums are:
X = A + B
Y = A + C
Z = B + C

Let’s add X and Y together

X + Y
= (A + B) + (A + C)
= 2A + B + C

This is (B+C) bigger than just 2A. Notice that Z = B + C so lets extend the first equation to include Z and the fact we’ll end up with 2A

(X + Y – Z) / 2
= [(A + B) + (A + C) – (B + C)] / 2
= [2A + B + C – B – C] / 2
= 2A / 2
= A

We can do this in any combination to get all the numbers, or just substitute A back into X and Y

(7 + 12 – 13) / 2 = 3
7 – 3 = 4
11 – 3 = 9

To check: 4 + 9 = 13

1. Ariel Marcy

Excellent algebraic solution, Martin!

2. Derek

Another algebraic solution:

Let the numbers be a, b and c, with a < b < c

a + b = 7 (equation 1)
a + c = 12 (equation 2)
b + c = 13 (equation 3)

There are many ways to solve for a, b and c. One example is to subtract equation 1 from equation 2:
(a + c) – (a + b) = 12 – 7
a + c – a – b = 5
c – b = 5 (equation 4)

Now add equation 3 and equation 4:
b + c + c – b = 13 + 5
2c = 18 (then divide both sides by 2)
c = 9

Substitute c = 9 in equations 2 and 3:
a + c = 12 (equation 2)
a + 9 = 12
a = 3

b + c = 13 (equation 3)
b + 9 = 13
b = 4

So, the numbers are 3, 4 and 9.

1. Ariel Marcy

Thanks, Derek! A creative and clearly written solution.

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