As your country's top spy, you must infiltrate the headquarters of the evil syndicate, find the secret control panel, and deactivate their death ray. But all you have to go on is the following information picked up by your surveillance team. The headquarters is a massive pyramid with a single room at the top level, two rooms on the next, and so on. The control panel is hidden behind a painting on the highest floor that can satisfy the following conditions: Each room has exactly three doors to other rooms on that floor, except the control panel room, which connects to only one, there are no hallways, and you can ignore stairs. Unfortunately, you don't have a floor plan, and you'll only have enough time to search a single floor before the alarm system reactivates. Can you figure out which floor the control room is on? Pause now to solve the riddle yourself. Answer in: 3 Answer in: 2 Answer in: 1 To solve this problem, we need to visualize it. For starters, we know that on the correct floor there's one room, let's call it room A, with one door to the control panel room, plus one door to room B, and one to C. So there must be at least four rooms, which we can represent as circles, drawing lines between them for the doorways. But once we connect rooms B and C, there are no other connections possible, so the fourth floor down from the top is out. We know the control panel has to be as high up as possible, so let's make our way down the pyramid. The fifth highest floor doesn't work either. We can figure that out by drawing it, but to be sure we haven't missed any possibilities, here's another way. Every door corresponds to a line in our graph that makes two rooms into neighbors. So in the end, there have to be an even number of neighbors no matter how many connections we make. On the fifth highest floor, to fulfill our starting conditions, we'd need four rooms with three neighbors each, plus the control panel room with one neighbor, which makes 13 total neighbors. Since that's an odd number, it's not possible, and, in fact, this also rules out every floor that has an odd number of rooms. So let's go one more floor down. When we draw out the rooms, low and behold, we can find an arrangement that works like this. Incidentally, the study of such visual models that show the connections and relationships between different objects is known as graph theory. In a basic graph, the circles representing the objects are known as nodes, while the connecting lines are called edges. Researchers studying such graphs ask questions like, "How far is this node from that one?" "How many edges does the most popular node have?" "Is there a route between these two nodes, and if so, how long is it?" Graphs like this are often used to map communication networks, but they can represent almost any kind of network, from transport connections within a city and social relationships among people, to chemical interactions between proteins or the spread of an epidemic through different locations. So, armed with these techniques, back to the pyramid. You avoid the guards and security cameras, infiltrate the sixth floor from the top, find the hidden panel, pull some conspicuous levers, and send the death ray crashing into the ocean. Now, time to solve the mystery of why your surveillance team always gives you cryptic information. Hi everybody. If you liked this riddle, try solving these two.
身為一個國家頂尖的間諜, 你須要潛入邪惡集團總部, 找到那機密的控制盤, 然後解除他們的死亡射線。 但你僅有以下情報 來繼續你的任務, 這是監控小組所收集來的。 總部是一個龐大的金字塔, 頂層有個單間房, 下一個樓層是兩間房, 以此類推。 控制盤藏在一幅畫的後面, 並在符合下列條件的最高樓層中: 每間房間有三個門 通到同樓層的不同房間, 除了控制房, 它只連到一個房間, 沒有走廊, 也不用考慮樓梯。 很不幸的是你沒有平面圖, 在警報系統恢復前, 你只有足夠的時間搜尋一層樓。 你能想出控制房是在哪一個樓層嗎? 先暫停一下自己來解這個謎, 三秒後揭曉 二秒後揭曉 一秒後揭曉 解決這個問題我們需要想像一下。 首先,我們知道對的樓層 有一間房間我們叫 A 房, 它有一個門通到控制房, 還有一個門去 B 房, 和一個門去 C 房, 所以至少會有四個房間。 我們可以用圓圈來表示, 它們之間畫線來表示門口。 一旦我們連接 B 房與 C 房, 就無法再有另一個連接的可能, 所以從樓頂算下來的第四層被剃除。 我們知道控制盤的樓層 必須越高越好, 所以我們從金字塔由上往下走。 第五高的層樓也不行, 我們可以透過畫圖來了解。 為了確認我們沒有漏掉任何的可能, 還有另一個方法。 在我們的圖中每個門都對應一條線, 使連接的兩房變成鄰居, 所以鄰居最後必須是雙數, 不管我們畫多少連接。 在第五高的樓層 若需符合我們一開始的條件, 那我們需要四個房間 它們各自有三個鄰居, 外加一個控制房只連到一個鄰居, 所以總共會是十三個鄰居。 因為十三是個單數,所以也不可能會是。 事實上只要是單數房的樓層 都可以排除。 我們再往下一層走, 當我們畫出這六個房間時, 你瞧,我們可以找到一個合乎答案的組合。 順道提一下,這種視覺化模型研究 顯示出不同對象之間的連接關係, 我們稱為 圖論 (graph theory)。 在一個基本的圖表中, 代表物體的圓稱為 結點 (node), 連接線叫作 邊 (edges)。 研究這類圖表的人員 會這樣提問題, 「這個結點離那個結點有多遠?」 「有最多連結的結點有多少條邊?」 「這兩個結點間有路徑嗎, 如果有,是多遠? 」 這種圖表常用來繪出通信網路, 但它們幾乎可以代表任何種類的網絡。 從一個城市裡的交通連結, 和人與人之間的社交關係, 到蛋白質之間的化學作用, 或疫情透過不同地點蔓延。 有了這些技巧後 我們返回金字塔, 你閃過了警衛與監視器, 潛入到第六高的樓層, 找到隱藏的控制盤, 拉下顯眼的控制桿, 將死亡射線墜毀海裡。 現在輪到解開這個謎: 為什麼你的監控小組 總是給你詭譎的情報? 大家好, 如果你喜歡這個謎題,可以再試試這兩個。