# Calculation puzzle 007

What is x?

76   26   24   24   30   x


There is a connection between these numbers. Based on that connection, find x. Possible answer is

18

Source: This question was asked in YTUYOS 2017 (Yildiz Technical University exam for International students in 2017) in booklet A, question no: 53.

• If there is only one possible answer, I would say: 18. – Florian F Jul 15 '20 at 21:04
• @FlorianF, can you elaborate, please? That answer is true, but how? Thanks. – garakchy Jul 15 '20 at 21:06
• add the digits of the first number 7+6=13 and multiply by 2 to get 26. Continue with that logic by multiplying with 3, 4, 5... I'm sure, this is according to FlorianF logic as well. – ThomasL Jul 15 '20 at 21:24
• @garakchy No downvote from me. The question looks good. It is just unusual that you give the answer straight away. – Florian F Jul 15 '20 at 21:34
• an upvote from my side, it's a good question, – ThomasL Jul 15 '20 at 21:39

Here is an explanation for 18 as a solution:

add the digits of the first number 7+6=13 and multiply by 2 to get 26. Continue with that logic by multiplying with 3, 4, 5. The last step gives the solution $$(3+0)*6=18$$

Consider the polynomial $$p(x) = (25x^4-242x^3+839x^2-1222x+912)/12$$. We get $$p(0)=76, p(1)=26, p(2)=24, p(3)=24, p(4)=30$$. The next number is $$p(5)=96$$.
this is a bit too generic answer which is not usually allowed, but I'm posting it here because the polynomial coefficients are not too ugly, and the result is an integer of reasonable magnitude (if the resulting answer was e.g. $$-\frac{10894}{349}$$ I would not even post it here). Moreover, since it is a technical university exam, it's reasonable to expect a somewhat technical solution to the problem.