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.
在仙境經歷了數次冒險之後, 愛麗絲再次來到 易怒的紅心皇后的宮庭。 她正要偷偷摸摸穿過花園時, 不小心聽見了國王和皇后在爭論。
“It’s quite simple,” says the queen. “64 is the same as 65, and that’s that.”
皇后說:「很簡單,」 「64 和 65 一樣,就這樣。」
Without thinking, Alice interjects. “Nonsense,” she says. “If 64 were the same as 65, then it would be 65 and not 64 at all.”
想也沒想,愛麗絲 就插嘴:「胡說。」 「如果 64 和 65 一樣, 那它就是 65,而不是 64 了。」
“What? How dare you!” the queen huffs. “I’ll prove it right now, and then it’s off with your head!”
皇后惱火:「什麼? 你好大的膽子!」 「我現在就能證明, 接著就要砍掉你的頭!」
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.
在能抗議之前, 愛麗絲就被拖向了 有兩個棋盤圖案的場地—— 8 乘以 8 的正方向 與 5 乘以 13 的長方形。 皇后拍手之後便來了 四個外表奇怪的士兵, 他們相鄰躺下,把第一個棋盤蓋住。 愛麗絲看見其中兩個士兵是梯形的, 斜邊以外的三個邊長為 5x5x3, 另外兩個士兵是長三角形, 斜邊以外的兩個邊長為 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."
「看,這是 64。」 皇后再次拍手。 撲克牌士兵爬起來,重新排列, 躺在第二個棋盤上。 「而這是 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?
愛麗絲很驚訝。 她肯定士兵從一個棋盤 到另一個棋盤時, 形狀大小都沒有變。 但皇后一定有以某種方式作弊, 從數學上來看絕對是如此。 愛麗絲能否想通問題在哪裏 ——在她輸掉之前?
Pause the video to figure it out yourself. Answer in 3.
請在此暫停解題。 答案即將公佈:三
Answer in 2
答案即將公佈:二
Answer in 1
答案即將公佈:一
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.
當情況看似對愛麗絲不利時, 她想起了她學過的幾何學, 再次去觀察併排躺下的梯形 和三角形士兵。 看起來,他們剛好蓋住半個長方形, 他們的邊緣形成 連接兩個對角的長邊。 如果真是如此,那麼他們 斜邊的斜率應該相同。 但當她使用數學上已證實的公式, 用高度除以長度 去計算這些斜率之後, 最妙的事情發生了。 梯形士兵的斜邊是 2 除以 5, 也就是 2/5 或 0.4。 然而,三角形士兵則是 3 除以 8, 斜率為 3/8 或 0.375。 兩個斜率完全不一樣! 皇后的守衛還沒能阻止愛麗絲, 她就喝下了縮小藥水,去看個清楚。 在三角形和梯形之間 的確有個小小的縫, 形成一個平行四邊形, 從棋盤的一端到另一端, 正好代表失蹤的那一塊 1x1 正方形。
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.
這些數字還有神奇之處, 它們都屬於費氏數列, 也就是說,每個數字 都是前兩個數字的加總。 在這裡,費氏數列的數字 有兩項特性十分重要: 第一,費氏數列中 任何一個數字的平方, 會比它在數列中前後兩個數字相乘 多 1 或少 1。 換言之,8 的平方 會比 5 乘以 13 少 1, 而 5 的平方會比 3 乘以 8 多 1。 第二, 費氏數列中,兩個相鄰 數字的比值會非常相近。 相近到它最終會收斂到黃金比率。 因為如此, 狡詐的皇室成員才能 做出看起來相似到可以唬人的斜邊。 事實上,紅心皇后 可以用費氏數列中的 任何四個連續數字 拼湊出類似的難題。 數字越大,不可能的事就越像真的。 但,引用路易斯卡羅的話—— 《愛麗絲夢遊仙境》的作者 及研究了這個謎題的高超數學家—— 我們不能去相信不可能的事。