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Here is a number sequence with a missing term:

$22$, $44$, $186$, $1458$, $11680$, $?$

The first thing I noticed is that if you take the first term ($22$) do $2 \times 2 = 4$ and that's the first digit of the second term; and if you do $2 + 2 = 4$ and that's the second digit of the second term. And then there's $44$, and $4 \times 4=16$ and $4 + 4 = 8$ and those are the digits in the third term, $186$.

Continuing this pattern, it works for $1458$ ($1 \times 4 \times 5 \times 8=160$ and $1+4+5+8=18$; $160$ and $18$ are the odd-positioned and even-positioned numbers in $11680$)

I noticed that the product and the sum fold together like that in all products.

I just need to know what number would replace the $?$. I'm thinking that it's $106$, because $1+1+6+8=16$ and $1 \times 1 \times 6 \times 8 = 0$

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closed as off-topic by F1Krazy, JonMark Perry, A. P., Glorfindel, ABcDexter Jan 6 at 18:29

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  • 2
    $\begingroup$ Please state where this problem comes from $\endgroup$ – Dr Xorile Jan 5 at 2:28
  • $\begingroup$ Please don't deface your post. If you want the question to be closed for violating our "no questions from ongoing competitions" rule, a simple comment will do. It's a bit too late now that you've already got two answers, though. $\endgroup$ – F1Krazy Jan 6 at 16:44
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I think $106$ is correct, though I do not think you ever put the full multiplication result in the middle, you are always alternating, but starting with one or the other based on where the result is in the series. Following your method above:

$22 \Rightarrow 2 + 2 = 4 $ and $2 \times 2 = 4 \Rightarrow 4 \ 4$ (order unclear)

$44 \Rightarrow 4 + 4 = 8 $ and $4 \times 4 = 16 \Rightarrow 1 \ 8 \ 6$ (start with multiplication result and alternate)

$186 \Rightarrow 1 + 8 + 6 = 15 $ and $1 \times 8 \times 6 = 48 \Rightarrow 1 \ 4 \ 5 \ 8$ (start with first digit of sum and alternate)

$1458 \Rightarrow 1 + 4 + 5 + 8 = 18 $ and $1 \times 4 \times 5 \times 8 = 160 \Rightarrow 1 \ 1 \ 6 \ 8 \ 0$ (start with first digit of multiplication and alternate)

And therefore, the next number should be:

$11680 \Rightarrow 1 + 1 + 6 + 8 + 0 = 16 $ and $1 \times 1 \times 6 \times 8 \times 0 = 0 \Rightarrow 1 \ 0 \ 6$ (start with first digit of sum and alternate)

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The product occupies the 1st, 3rd, 5th etc digit from the right, the sum occupies the 2nd, 4th, 6th etc. With a product of 0 and a sum of 16 we have _0_0 and 1_6_ so the next number should be 1060, followed by 70 as the final number in the sequence.

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