.

I went ahead and tried a number of possible continuations of this sequence, leaving out different numbers.

The Lagrange polynomial utility was never able to predict any one of the missing numbers, any which way.

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I input 1,2,4 for x, and 1,12,1 for y. This means our original series with the 3rd number missing: 1,12,...,1

The calculator provided P(x) = 12 which translates to their answer being **1, 12, 12, 1** --- but it should be 1,12,1,1

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Next, I input 1,2,5 for x, and 1,12,1 for y. This would be for our series with the 3rd and 4th numbers missing: 1,12,...,...,1

The calc gave P(3) = 15.67 and P(4) = 12 which means their answer is **1, 12, 15.67, 12, 1 --- **should be 1,12,1,1,1

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Next, I input 1,2,6 for x, and 1,12,2 for y. This is for our series with the 3rd, 4th and 5th missing: 1,12,...,...,...,2

The calc gave P(3) = 17.60, P(4) = 17.80, P(5) = 12.60 or, **1, 12, 17.60, 17.80, 12.60, 2** --- should be 1,12,1,1,1,2

.

When I left out the first number things got strange.

2,3,5 for x, 12,1,1 for y outputs **30.33, 12, 1, -2.67, 12** --- instead of 1,12,1,1,1,2

2,3,4,6 for x, 12,1,1,2 for y outputs **42, 12, 1, 1, 2, 4 **

.

This formula is utterly useless on this puzzle. The formula clearly has zero relation to the puzzle key.

Sometimes relying on pure mathematics puts you everywhere EXCEPT in the right place.

This polynomial calculator is entirely unable to anticipate any one of the numbers in the subject series!