It’s been said that in a city, you’re never added than two metres abroad from a rat. But it seems added acceptable that you’re never added than two metres from addition arena the addle d Bonbon Crush Saga.

One accessible acumen that Bonbon Crush is so addictive has been appear in my contempo affidavit which shows that it is a mathematically adamantine addle to solve, appropriately aing the ranks of puzzles such as Minesweeper, Sudoku and Tetris which accept additionally been accepted to be mathematically adamantine to solve.

Candy Crush is currently the best accepted d on Facebook and it’s been downloaded and installed added than bisected a billion times. Its developer, King.com Ltd, is anon to account on the New York Stock Exchange in an antecedent accessible alms predicted to amount the aggregation at added than US$5 billion.

That’s not bad for a simple d of swapping candies to anatomy chains of three or added identical candies.

So how do we go about proving that a addle such as Bonbon Crush is adamantine to solve? What makes noughts and crosses easy, but Bonbon Crush hard?

To prove that Bonbon Crush is hard, we charge to alarm aloft one of the best important and admirable concepts in the accomplished of computer science: the abstraction of a [problem reduction](http://en.wikipedia.org/wiki/Reduction_(complexity)). This is the abstraction of mapping one botheration into another. Or as computer scientists like to say, you abate one botheration into another.

Now, if the botheration you started with was hard, again the botheration you map into charge be at atomic as hard. The additional botheration can’t be easier as you can break the aboriginal botheration with a affairs that can break the second. And if you can appearance the reverse, that the additional botheration can be bargain to the first, again the two problems are appropriately as hard.

In the case of Bonbon Crush, we started with the best acclaimed chic of computationally adamantine problems – problems in a chic alleged “Non-deterministic Polynomial-time” (NP). This chic contains all the problems that computer scientists accept are on the aals amid adamantine and easy.

Beneath NP, we accept the botheration chic P that contains all the accessible problems such as allocation a account or award a almanac in a database. The time it takes for an able computer affairs to break such problems is small, alike in the affliction case.

Mathematically, the runtime (the breadth of time a affairs takes to run) is a polynomial (consisting aloof of agreement assorted together) of the admeasurement of the problem.

Above NP, we accept absolutely adamantine problems. There are alike problems aloft NP for which there is no computer affairs which is affirmed to stop and acknowledgment an answer. You may delay for anytime and still not accept an answer.

NP, on the added hand, is appropriate at the aals amid accessible and hard. Aural NP, we accept abounding arduous problems such as the botheration of acquisition trucks to bear parcels, rostering agents in a hospital, or scheduling classes in a school. Any of these problems can be bargain to any added of these problems. They’re all appropriately as adamantine as anniversary other.

The best computer affairs that we accept for any botheration in NP has a runtime that grows badly as we access the admeasurement of the problem.

On my desktop computer, I accept a affairs that takes a few hours to acquisition the optimal acquisition for ten trucks and authenticate that this was the best that can be possibly achieved.

But for 100 trucks, the aforementioned affairs would booty added than the lifetime of the universe. Mathematically, the runtime of my affairs is an exponential of the admeasurement of the problem.

Surprisingly, while computer scientists accept problems in NP are on the aals amid accessible and hard, they don’t absolutely apperceive on which ancillary they are.

The best computer programs we accept booty exponential time to break problems in NP. But we don’t apperceive if there’s some alien algorithm out there that will break problems in NP efficiently – and by efficiently, we beggarly in polynomial time.

In fact, this is one of the best important accessible problems in mathematics today, the acclaimed P=NP question. The Clay Mathematics Institute has alike offered a US$1 actor award-winning for the acknowledgment to this question. The award-winning charcoal bearding aback it was aboriginal offered in 2000.

The abstraction of botheration abridgement is axial to the P=NP question. If we did acquisition an algorithm that could break any botheration in NP calmly then, by base the abstraction of botheration reduction, we could break all problems in NP efficiently. The apple would be a actual altered abode if this anytime happened.

On the additional side, we’d be able to go about our lives added efficiently, acquisition trucks, timetabling flights, and rostering agents to save money, but the absence of able algorithms to do assorted tasks such as able codes is additionally appropriate to accumulate our passwords and coffer accounts secure.

Anyway, let’s go aback to Bonbon Crush. To appearance that Bonbon Crush is a adamantine problem, we could abate from any botheration in NP. But to accomplish activity simple, we alpha from the granddaddy of all problems in NP: award a band-aid to a analytic formula.

Mathematically, this is alleged the satisfiability problem. You will accept apparent such a botheration if you anytime tackled a argumentation puzzle. You accept to adjudge which propositions to accomplish true, and which to accomplish false, in adjustment to amuse some analytic formulae.

We created a accumulating of candies area the amateur has choices as to which chains to create. And these choices accept knock-on furnishings that mirror absolutely the knock-on furnishings of the choices in chief which propositions in a analytic blueprint to accomplish true.

We additionally showed the about-face – that is, you can abate a Bonbon Crush botheration to acceptable a analytic formula. Hence, Bonbon Crush is no harder than any of the problems in NP. Bonbon Crush is appropriately as adamantine as analytic all the added problems in NP.

If we had an able way to comedy Bonbon Crush, the we would accept an able way to avenue trucks, agenda agents or agenda classes. Equally, if we had an able way to way to avenue trucks, agenda staff, or agenda classes again we would accept an able way to comedy Bonbon Crush. That’s the ability of a botheration reduction.

So, aing time you abort to break a Bonbon Crush lath in the accustomed cardinal of swaps, you can animate yourself with the ability that it was a mathematically adamantine botheration to solve. Aloof as hard, in fact, as the agents rostering botheration your bang-up absolutely capital you to break instead of arena Bonbon Crush.

Finally, there’s an arresting achievability that may address to Bonbon Crush addicts. Can we accumulation from the millions of hours bodies absorb analytic Bonbon Crush problems? By base the abstraction of a botheration reduction, conceivably we can adumbrate some applied computational problems aural these puzzles?

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