After many adventures in Wonderland, Alice has once again found herself in the court of the temperamental Queen of Hearts. She’s about to pass through the garden undetected, when she overhears the king and queen arguing.
Nakon mnogih pustolovina u Zemlji čudesa, Alisa se ponovno našla na dvoru temperamentne Kraljice Srca. Dok pokušava neopaženo proći kroz vrt, začuje kralja i kraljicu kako se prepiru.
“It’s quite simple,” says the queen. “64 is the same as 65, and that’s that.”
"Vrlo jednostavno", kaže kraljica. "64 je isto što i 65, i to je to."
Without thinking, Alice interjects. “Nonsense,” she says. “If 64 were the same as 65, then it would be 65 and not 64 at all.”
Bez razmišljanja, Alisa upadne u riječ. "Glupost. Da je 64 isto što i 65, tada bi bilo 65, a nikako ne 64."
“What? How dare you!” the queen huffs. “I’ll prove it right now, and then it’s off with your head!”
"Što? Kako se usuđuješ!" plane kraljica. "To ću odmah dokazati, a onda ti ode glava!"
Before she can protest, Alice is dragged toward a field with two chessboard patterns— an 8 by 8 square and a 5 by 13 rectangle. As the queen claps her hands, four odd-looking soldiers approach and lie down next to each other, covering the first chessboard. Alice sees that two of them are trapezoids with non-diagonal sides measuring 5x5x3, while the other two are long triangles with non-diagonal sides measuring 8x3.
Alisa se nije stigla ni pobuniti, i već je odvučena na travnjak s dvije šahovske ploče -- s kvadratom 8 x 8 i pravokutnikom 5 x 13. Dok kraljica plješće rukama, četiri stražara čudnog izgleda prilaze i legnu jedan do drugoga, prekrivajući prvu šahovnicu. Alisa vidi da dvojica od njih, trapezoidi nedijagonalnih stranica, mjere 5x5x3, dok su druga dva dugački trokuti s nedijagonalnim stranicama mjera 8x3.
“See, this is 64.” The queen claps her hands again. The card soldiers get up, rearrange themselves, and lie down atop the second chessboard. “And that is 65."
"Vidiš, ovo je 64." Kraljica ponovno pljesne rukama. Stražari karte ustaju, preslože se i legnu na drugu šahovsku ploču. "A to je 65."
Alice gasps. She’s certain the soldiers didn’t change size or shape moving from one board to the other. But it’s a mathematical certainty that the queen must be cheating somehow. Can Alice wrap her head around what’s wrong— before she loses it?
Alisa je zgranuta. Stražari sigurno nisu promijenili veličinu ni oblik prelazeći s jedne ploče na drugu. Ali matematički je sigurno da kraljica mora nekako varati. Može li Alisa odgonetnuti što nije u redu - prije no što izgubi glavu?
Pause the video to figure it out yourself. Answer in 3.
Pauzirajte video da biste sami shvatili. Odgovor u 3,
Answer in 2
2,
Answer in 1
1.
Just as things aren’t looking too good for Alice, she remembers her geometry, and looks again at the trapezoid and triangle soldier lying next to each other. They look like they cover exactly half of the rectangle, their edges forming one long line running from corner to corner. If that’s true, then the slopes of their diagonal sides should be the same. But when she calculates these slopes using the tried and true formula "rise over run," a most curious thing happens. The trapezoid soldier’s diagonal side goes up 2 and over 5, giving it a slope of two fifths, or 0.4. The triangle soldier’s diagonal, however, goes up 3 and over 8, making its slope three eights, or 0.375. They’re not the same at all! Before the queen’s guards can stop her, Alice drinks a bit of her shrinking potion to go in for a closer look. Sure enough, there’s a miniscule gap between the triangles and trapezoids, forming a parallelogram that stretches the entire length of the board and accounts for the missing square.
I baš kad stvari ne idu Alisi u prilog, ona se sjeti svoje geometrije, i ponovno pogleda trapezoidnog i trokutastog stražara polegnute jedan uz drugog. Čini se kao da pokrivaju točno pola pravokutnika, a njihove stranice tvore jednu dugu crtu koja prolazi od kuta do kuta. Ako je to istina, tada bi nagibi njihovih dijagonalnih stranica trebali biti jednaki. Ali kad izračuna te nagibe koristeći provjerenu formulu "rast kroz put", dogodi se najčudnovatija stvar. Dijagonalna stranica trapezoidnog stražara raste za 2 i pomiče se za 5, dajući nagib od dvije petine ili 0,4. Dijagonala trokutnog stražara, međutim, raste za 3 i pomiče se za 8, dajući nagib od tri osmine ili 0,375. Oni uopće nisu isti! Prije nego što je stražari mogu zaustaviti, Alisa otpije malo napitka za smanjivanje da to pobliže pogleda. Svakako, postoji mali razmak između trokuta i trapeza, i tvori paralelogram koji se proteže cijelom duljinom ploče i vrijedi za kvadrat koji nedostaje.
There’s something even more curious about these numbers: they’re all part of the Fibonacci series, where each number is the sum of the two preceding ones. Fibonacci numbers have two properties that factor in here: first, squaring a Fibonacci number gives you a value that’s one more or one less than the product of the Fibonacci numbers on either side of it. In other words, 8 squared is one less than 5 times 13, while 5 squared is one more than 3 times 8. And second, the ratio between successive Fibonacci numbers is quite similar. So similar, in fact, that it eventually converges on the golden ratio. That’s what allows devious royals to construct slopes that look deceptively similar. In fact, the Queen of Hearts could cobble together an analogous conundrum out of any four consecutive Fibonacci numbers. The higher they go, the more it seems like the impossible is true. But in the words of Lewis Carroll— author of Alice in Wonderland and an accomplished mathematician who studied this very puzzle— one can’t believe impossible things.
Još je nešto čudnovatije u vezi s tim brojevima: svi su oni dio Fibonaccijevog niza, gdje je svaki sljedeći broj zbroj dvaju prethodnih. Fibonaccijevi brojevi imaju dva svojstva koja ovdje igraju ulogu: prvo, kvadrat Fibonaccijevog broja daje vam vrijednost koja je za jedan više ili jedan manje različiita od umnoška Fibonaccijevih brojeva s obje njegove strane. Drugim riječima, 8 na kvadrat je jedan manje od 5 puta 13, dok je 5 na kvadrat za jedan više od 3 puta 8. I drugo, omjer između uzastopnih Fibonaccijevih brojeva prilično je sličan. Toliko sličan da se na kraju pretapa u zlatni rez. To je ono što omogućuje nepoštenim vladarima da grade varljivo slične nagibe. Zapravo, Kraljica Srca mogla bi skovati sličnu zagonetku od bilo koja četiri uzastopna Fibonaccijeva broja. Što su ti brojevi veći, to ono nemoguće više liči na istinu. Ali riječima Lewisa Carrolla -- autora knjige Alisa u Zemlji čudesa i vrsnog matematičara koji je proučavao upravo ovu zagonetku -- ne može se vjerovati u nemoguće stvari.