Barbara sent in this doozy. I have yet to figure it out.

What is the missing number?

6, 9, 27, 54, ?, 2241

Anyone have any ideas?

When you’re nearly comatose from over consumption of Thanksgiving’s finest, figure this out to give your brain some exercise.

Hmmm … 6 X 9 = 54, 27 X 2 = 54 … wait, do you know the answer, or have access to it? How will we know? Is it 83? ‘Cause 27 X 83 = 2241? That doesn’t seem to fit a pattern or make much sense, but hey, maybe worth a shot! 243? ‘Cause 9 X 27 = 243? …

I don’t have the answer, but a reasonably generic formula that outputs any number in the series should be satisfying enough.

Here are my notes from various attempts at finding a pattern.

2 x 3 = 6

3 x 3 = 9

9 x 3 = 27

18 x 3 = 54

747 x 3 = 2241

Six

Nine

Twenty-seven

Fifty-four

Two thousand two hundred forty one

Factors

6 = 1,2,3,6

9 = 1,3,9

27 = 1,3,9,27

54 = 1,2,3,6,9,18,27,54

2241 = 1,3,9,27,83,249,747,2241

Addition

6 (+3 = 9)

9 (+18 = 27)

27 (+27 = 54)

54 (+?)

Multiplication

6 (x1.5 = 9)

9 (x3 = 27)

27 (x2 = 54)

54 (x?)

Just heard back from Barbara that she doesn’t know the answer, but it’s a three digit number that starts with 65. Since all the other numbers are divisible by three, I’m guessing it’s either 651, 654 or 657 (217, 218 and 219 x3).

There are no multiples of 54 that fall within the range of 650-659.

I think I’ve got it…(although to be honest, nothing clicked until I read one of the above posts).

6

9

27

54

X

2241

The pattern I found is that the sum of each two consecutive numbers is equal to the previous number squared (Although I only have proof of this for a few terms). For example:

9 + 27 = 6^2

27 + 54 = 9^2

54 + X = 27^2

X + 2241 = 54^2

1) 54 + X = 729

X = 675

2) X + 2241 = 2916

X = 675

I get the same value for X both times, which supports this hypothesis o_O

@Parris Nice work. That’s got to be the answer. Barbara was mistaken – the answer starts with 67, not 65.

6 9 27 54 _ 2241

Ans:-675

6*6-9=27

9*9-27=54

27*27-54=675

54*54-675=2241

I think I have it, except for the 6, if you add the digits in each number they come to total of 9, eg 2 + 7 = 9, 5 + 4 = 9, 2+2+4+1 = 9, so missing number could be 81 or 621. What do you think?

@ann But what about the 6? The solution should work for all numbers in the series.

How about turning the 6 upside down!

@ann You get points for creativity, but Parris’ mathematical solution is far more satisfying.