Difficulty: Tricky

Spencer has a set of five numbers arranged from smallest to largest: 1, 3, 3, 5, 8.

He calculates some stats about this set of numbers including the middle number, the most common number and the average.

Here are Spencer’s stats:

• Middle number = 3
• Most common number = 3 (and only 3)
• Average = 4

Can you find a different set of 5 positive whole numbers with the same stats? Post your solutions in the comments! Remember that the set must be ordered from smallest to largest.

Don’t know how to find an average? The average is calculated by adding up all five numbers and then dividing by the number of numbers in the set (in this case, 5). For Spencer’s numbers:

(1 + 3 + 3 + 5 + 8) ÷ 5 = 4

Need a hint?

Make sure you pay attention to all 3 stats!

The most common number can be tricky – make sure you don’t accidentally have 2 different most common numbers.

To calculate the average, you add up all the numbers, and divide by how many there are. We can use this formula to get a big clue.

(Spencer’s average) = (all the numbers added up) ÷ (number of numbers)

4 = (all the numbers added up) ÷ 5

What number divided by 5 equals 4? 20! So, the rule we get from the average is that all the numbers in our set need to add up to 20.

Brainteaser answer

Yes, it is possible to have a different set of 5 numbers with the same stats. Here are a few possible solutions:

• 1, 2, 3, 3, 11
• 2, 3, 3, 3, 9
• 2, 3, 3, 4, 8

Each of Spencer’s stats gives a different clue about our solution.

First, the middle number = 3. This means that when the numbers are arranged from smallest to largest, the middle number has to be a 3.

Second, the most common number = 3. This means we need to have at least two 3’s. We also learn that the other numbers need to occur fewer times than 3 does.

Third, the average = 4. To get the average, you first need to add up all the numbers and divide by the number of numbers in the set. We know from the problem that our set has 5 numbers. To match Spencer’s stats, we know that the average must equal 4.

Average = (all the numbers added up) ÷ (number of numbers) = 4

Average = (all the numbers added up) ÷ 5 = 4

Now, we are just missing the top part of the fraction. What number divided by 5 equals 4? 20! This means the numbers in our set must add up to 20 to have the same average as Spencer’s set.

Now, you can use these rules to pick different sets of numbers with the same stats.
Start with 3 in the middle:
_,_ , 3_ ,_

Now add at least one more 3, for this example, we’ll add one to the left:
_, 3, 3,_ , _

Now choose numbers that, in total, add up to 20. Only one number can be less than 3, for this example, we’ll choose the number 2:
2, 3, 3,_ ,_

So far, the numbers we’ve chosen add up to 8. This means the last two must add up to 12. We can’t choose 6 and 6 because that would mean our new set of numbers has two most common numbers, 3 and 6, making it different from Spencer’s stats. So, choose two other numbers that add up to 12 and are both larger than or equal to 3. Here’s one solution:
2, 3, 3, 4, 8

Did you find a different solution? Post it in the comments and tell us how you got there!

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