# Lalozik Pattern

Vasily Lalozik is a professional cryptographer and develops new security schemes for banks. But he has one problem: David Banes.

The American Bank is the largest bank in the world; Banes has his sights steered towards it. Lalozik has created a very tough security scheme for the bank, which is considered almost impossible to crack by any outsider. The scheme is very simple and straightforward. The man wanting to enter the money container has to input the answer of a pattern. The pattern's technique of solving has been disclosed only to the owner and the respective manager.

On June 9th, Banes managed to get inside the bank and has only one remaining task: to crack the puzzle and run away with approximately 4 billion dollars.

The pattern is this:

192-30-55-86-43-22-79-3-?

What number comes next in the series?

• I feel that if you expect us to solve this, Lalozik must not be very good at his job. – Ian MacDonald Apr 3 '15 at 21:19
• Also, all that setup for a simple sequence puzzle... – Joe Z. Apr 3 '15 at 22:13
• This puzzle is now the top Google hit for "Lazolik". Well done on inventing a new name! :-p – Rand al'Thor Apr 3 '15 at 22:57
• Do not question God. It doesn't end well. – Aggie Kidd Apr 3 '15 at 23:12
• The United American Bank opened its doors on June 9, 2003. Jim Clark, infamous American bank robber, died on June 9, 1974. Coincidence?? – Ben Frankel Apr 7 '15 at 11:04

tl;dr: most likely not solvable with the information given.

The next number is $-642$ and the pattern is formed by

$\frac{1}{18}\cdot{}x^7 - \frac{1411}{720}\cdot{}x^6 + \frac{19441}{720}\cdot{}x^5 - \frac{26609}{144}\cdot{}x^4 + \frac{94175}{144}\cdot{}x^3 - \frac{49714}{45}\cdot{}x^2 + \frac{12509}{20}\cdot{}x^1 +177$.

Well, probably not: it isn't stated in the question, but the solution presumably is a positive integer (though regression analysis suggests that it has to be negative).

The most straightforward solution actually is simple addition:

$510 = 192+30+55+86+43+22+79+3$

More serious considerations:

I shall be assuming that this is purely mathematical, i.e. not relying on trivia like the character's names, the reciptionist's birthday or the arrangement of buttons on the input pad. Furthermore people go into finance because they are bad at maths, so only elementary operations are permitted and the numbers are used in the order they show up in (and not reverse or sorted ascending or descending, alternating front/back or some non-derivable order). Nor shall there be skips in the sequence (e.g. taking only the prime-numbered elements of the sequence generated).

If I were Lalozik I would have the sequence of numbers change every time the safe is attempted to be opened: while the rule for generation and solution would remain the same, the seed for generation would change randomly, hence one couldn't predict the next number without knowing the algorithm. However, this is obviously not the case, since otherwise Banes would have staked out the bank beforehand and seen the patterns and possibly the solutions to previous times the safe was opened, so those sequences would be provided to us for solution.

For it to be possible to derive a solution by looking at the list of numbers with such restricted information, the algorithm has to repeat: for example, for $1, 2, 3, 5, 8$ one would guess the fibonacci numbers where $x_{n+2}=x_{n+1}+x_{n}$, but this cannot be derived if only $1, 2$ are given since $3$ would be the very first number where the formula applies. Indeed, even $1, 2, 3$ doesn't suffice: there need at least two repeats of the application of the formula within the sequence, one to establish the pattern and one to affirm it.

Otherwise if someone came up with a different solution one couldn't dismiss it: if there is no repeating pattern then straightforward addition is the simplest solution. Hence at most 7 of the numbers can be used to derive the solution, the 8th must serve for confirmation - the fewer needed the more plausible the rule becomes.

In fact, it can be easily shown (google "common difference") that in the present case there exists no polynomial that can generate these numbers in a manner that permits the solution to be confirmed. Of course, this still allows any algorithm that is more complex than a polynomial (e.g. involving floor functions or cosines) - though then the claim that the scheme is "very simple and straightforward" is misleading at best.

• I'd like to upvote for "people go into finance because they are bad at maths" and downvote for the -642 answer. So I won't vote on this at all :-) – Rand al'Thor Apr 12 '15 at 9:59
• Using a curve-fitting technique is definitely not the way to uncover the sequence of an encryption key. If you had added in some stuff about the entropy though, then that might've made sense. – Allan Apr 12 '15 at 19:05
• @Allan: the initial curve-fitting was a joke. The number sequence isn't necessarily a cryptogram. In either case, if it was a cryptogram it appears to be one where the key is too long and the plaintext too short to permit solution. The entropy cannot be reliably calculated for such a short snippet. Hence I restricted myself to pointing out that a process that is both simple (in the sense of that a bank manager could calculate it in his head) and non-random cannot yield that sequence. – user66554 Apr 13 '15 at 6:57

I think the next number in the sequence is: 99

The numbers currently add up to 510. 510+99 is 609 (the date)

Or possibly: 320

By taking the numbers at positions 1,9,2 after the 192

• Maybe, but if so, then this really isn't a pattern, but only a list of numbers. The question does specify that it is a pattern, so I think that means that each number (after the first one usually) could be figured out by looking at the numbers to the left of it. – JLee Apr 12 '15 at 19:04
• Perhaps that is lazolik's security through obscurity? :D – Andrew Smith Apr 12 '15 at 19:06
• The word pattern is even in the title. Questions like these are so open-ended that there are infinite valid answers really. So, I guess I am just looking for an answer that seems the easiest and most "cool", and then hoping that Lalozik's idea of "cool" is the same as mine. haha – JLee Apr 12 '15 at 19:09
• Yea, same thoughts here, that's why I tried the date. It could just be a pattern of random numbers that add up to the date, or a pattern of random numbers deciphered using the first number as the key – Andrew Smith Apr 12 '15 at 19:12
• is "random pattern" an oxymoron? if not, then what does pattern mean? – JLee Apr 12 '15 at 19:16

192-30-55-86-43-22-79-3 = -126

So i'd guess 126

Im still thinking that you would only say June 9th if there was a reason for us to know that it was a specific date Same goes for the amount of money thats currently in the vault. Anyone outsider would know the date, but only the bank manager and owner would know the amount of money in the bank on that date.

So maybe the answer is 4?