Your favorite band is great at playing music, but not so great at being organized. They keep misplacing their instruments on tour, and it's driving their manager mad. On the day of the big concert, the band wakes up to find themselves tied up in a windowless, soundproof practice room. Their manager explains what's happening. Outside, there are ten large boxes. Each contains one of your instruments, but don't be fooled by the pictures - they've been randomly placed. I'm going to let you out one at a time. While you're outside, you can look inside any five boxes before security takes you back to the tour bus. You can't touch the instruments or in any way communicate what you find to the others. No marking the boxes, shouting, nothing. If each one of you can find your own instrument, then you can play tonight. Otherwise, the label is dropping you. You have three minutes to think about it before we start. The band is in despair. After all, each musician only has a 50% chance of finding their instrument by picking five random boxes. And the chances that all ten will succeed are even lower - just 1 in 1024. But suddenly, the drummer comes up with a valid strategy that has a better than 35% chance of working. Can you figure out what it was? Pause the video on the next screen if you want to figure it out for yourself! Answer in: 3 Answer in: 2 Answer in: 1 Here's what the drummer said: Everyone first open the box with the picture of your instrument. If your instrument is inside, you're done. Otherwise, look at whatever's in there, and then open the box with that picture on it. Keep going that way until you find your instrument. The bandmates are skeptical, but amazingly enough, they all find what they need. And a few hours later, they're playing to thousands of adoring fans. So why did the drummer's strategy work? Each musician follows a linked sequence that starts with the box whose outside matches their instrument and ends with the box actually containing it. Note that if they kept going, that would lead them back to the start, so this is a loop. For example, if the boxes are arranged like so, the singer would open the first box to find the drums, go to the eighth box to find the bass, and find her microphone in the third box, which would point back to the first. This works much better than random guessing because by starting with the box with the picture of their instrument, each musician restricts their search to the loop that contains their instrument, and there are decent odds, about 35%, that all of the loops will be of length five or less. How do we calculate those odds? For the sake of simplicity, we'll demonstrate with a simplified case, four instruments and no more than two guesses allowed for each musician. Let's start by finding the odds of failure, the chance that someone will need to open three or four boxes before they find their instrument. There are six distinct four-box loops. One fun way to count them is to make a square, put an instrument at each corner, and draw the diagonals. See how many unique loops you can find, and keep in mind that these two are considered the same, they just start at different points. These two, however, are different. We can visualize the eight distinct three-box loops using triangles. You'll find four possible triangles depending on which instrument you leave out, and two distinct paths on each. So of the 24 possible combinations of boxes, there are 14 that lead to faliure, and ten that result in success. That computational strategy works for any even number of musicians, but if you want a shortcut, it generalizes to a handy equation. Plug in ten musicians, and we get odds of about 35%. What if there were 1,000 musicians? 1,000,000? As n increases, the odds approach about 30%. Not a guarantee, but with a bit of musician's luck, it's far from hopeless. Hi everybody, if you liked this riddle, try solving these two.
Tvoj omiljeni bend je odličan u sviranju, ali ne tako dobar kada se treba organizirati. Stalno zameću svoje instrumente na turneji, a to izluđuje njihovog menadžera. Na dan velikog koncerta, članovi benda probude se zatočeni u zvučno izoliranoj sobi za probe bez prozora. Njihov menadžer objasni im što se događa. Vani je deset velikih kutija. Svaka sadrži jedan od vaših instrumenata, ali ne dajte da vas slike zavaraju; postavljene su nasumično. Puštat ću vas jednog po jednog. Dok ste vani, možete pogledati u bilo kojih pet kutija, prije nego vas zaštitari vrate u autobus. Ne smijete dirati instrumente ili na neki način drugima pokušati reći što ste našli. Nema označavanja kutija, vikanja, ničega. Ako svatko od vas uspije naći svoj instrument, večeras možete svirati. U suprotnom, diskografska kuća raskida ugovor s vama. Imate tri minute za razmišljanje prije početka. Bend je u očaju. Svaki glazbenik ima samo 50% šanse za pronalaženje svog instrumenta, bude li birao pet nasumičnih kutija. Šanse da svi uspiju čak su i manje; samo 1 : 1024. Iznenada, bubnjar je iznio razumnu strategiju koja ima više od 35% šanse za uspjeh. Možeš li shvatiti što je predložio? Pauziraj video na idućem zaslonu ako želiš sam riješiti zagonetku. Odgovor za: 3 Odgovor za: 2 Odgovor za: 1 Bubnjar je rekao sljedeće: Svatko neka prvo otvori kutiju sa slikom svojeg instrumenta. Ako je unutra tvoj instrument, gotov si. U suprotnom, pogledaj što je u kutiji i zatim otvori kutiju koja ima sliku tog instrumenta. Nastavi dok ne nađeš svoj instrument. Članovi benda su skeptični, no, začuđujuće, svi nađu što trebaju. Nekoliko sati kasnije, sviraju za tisuće svojih obožavatelja. Zašto je bubnjarova strategija djelovala? Svaki glazbenik prati povezani niz koji počinje kutijom čija slika odgovara njihovom instrumentu i završava kutijom koja ga sadrži. Opazi da bi se vratili na početak kada bi nastavili, dakle, ovo je petlja. Na primjer, da su kutije ovako poslagane, pjevačica bi otvorila prvu kutiju i našla bubnjeve, otišla do osme kutije i našla bas gitaru i našla mikrofon u trećoj kutiji, koja bi je vodila natrag do prve. Ovo djeluje puno bolje od nasumičnog pogađanja, jer ako započne kutijom sa slikom svojeg instrumenta, svaki glazbenik ograničava potragu na petlju koja sadrži njihov instrument i postoji pristojna mogućnost, oko 35%, da će sve petlje biti duge pet ili manje koraka. Kako smo izračunali tu mogućnost? U svrhu jednostavnosti, demonstrirat ćemo na pojednostavljenom slučaju; četiri instrumenta i ne više od dva pokušaja po glazbeniku. Počnimo pronalaženjem mogućnosti neuspjeha, slučajem da će netko morati otvoriti tri ili četiri kutije, prije nego pronađe svoj instrument. Postoji šest jedinstvenih petlji s četiri kutije. Jedan zabavan način da ih izbrojiš jest da napraviš kvadrat. Stavi po jedan instrument u svaki kut i povuci dijagonale. Vidi koliko jedinstvenih petlji možeš naći, ali imaj na umu da se ove dvije smatraju jednakima, samo kreću iz različite točke. Međutim, ove su dvije različite. Možemo si predočiti osam različitih petlji s tri kutije koristeći trokute. Naći ćeš četiri moguća trokuta, ovisno o instrumentu kojeg izostaviš, i dva različita puta na svakom. Dakle, od 24 mogućih kombinacija kutija, 14 ih vodi k neuspjehu, a 10 rezultira uspjehom. Ta računska strategija vrijedi za sve parne brojeve glazbenika, no, ako želiš kraticu, može se svesti na korisnu jednadžbu. Uključi deset glazbenika i dobit ćemo šanse od oko 35%. Što kada bismo imali tisuću glazbenika? Milijun? Kako se n povećava, šanse se približavaju 30-ak posto. Nije garancija za uspjeh, no uz malo sreće, stvar je daleko od beznadne. Ako vam se svidjela ova zagonetka, pokušajte riješiti ove dvije.