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.
Budući da ste najbolji špijun u državi, morate da se ušunjate u sedište zlog udruženja, nađete tajnu kontrolnu tablu, i deaktivirate njihov zrak smrti. No, sve što znate o tome su sledeće informacije koje je prikupio vaš tim za prismotru. Sedište je ogromna piramida sa samo jednom prostorijom na najvišem nivou, dve prostorije na sledećem nivou, i tako dalje. Kontrolna tabla je sakrivena iza slike na najvišem spratu, koji zadovoljava sledeće uslove: Svaka prostorija ima tačno troje vrata koje povezuju druge prostorije na tom spratu, osim prostorije sa kontrolnom tablom, koja je povezana samo sa jednim vratima, nema hodnika, stepenice se ne računaju. Nažalost, nemate plan spratova, i imate dovoljno vremena da pretražite samo jedan sprat pre nego što se sistem za uzbunu ponovo aktivira. Možete li da odgonetnete na kom spratu se nalazi kontrolna soba? Pauzirajte sada da sami rešite zagonetku. Odgovor za: 3 Odgovor za: 2 Odgovor za: 1 Da rešimo ovaj problem moramo da pribegnemo vizualizaciji. Za početak, znamo da je na pravom spratu jedna soba, nazovimo je soba A, sa jednim vratima koje vode do kontrolne sobe, uz jedna vrata koja vode do sobe B, i jedna koja vode do sobe C. Stoga, mora da postoji najmanje četiri sobe, koje možemo predstaviti u vidu krugova, uz linije nacrtane između njih koje predstavljaju dovratke. No kada povežemo sobe B i C, nema nijedne druge moguće povezanosti, stoga četvrti sprat od gore otpada sa spiska. Znamo da kontrolna tabla mora da bude na što višem mestu moguće, stoga je najbolje da se spuštamo kroz piramidu. peti najviši sprat takođe ne odgovara. To možemo da shvatimo kada ga nacrtamo, no, da budemo sigurni da nismo nešto propustili, evo još jednog načina. Svaka vrata odgovaraju jednoj liniji u našem grafikonu koji predstavlja dve povezane sobe. tako da, na kraju, mora postojati paran broj povezanih soba bez obzira na to koliko povezanih soba označimo. Na petom najvišem spratu, da ispunimo početne uslove, trebalo bi nam četiri sobe sa još tri povezane sobe, uz kontrolnu sobu koja bi imala još jednu povezanu sobu, što ukupno čini 13 povezanih soba. Budući da je to neparan broj nije moguće, zapravo, ovo takođe eliminiše svaki sprat koji ima neparan broj soba. Stoga idemo još jedan sprat niže. Kada nacrtamo sobe, kako ispada, dobijamo raspored koji izgleda ovako. Sasvim slučajno, ispitivanje ovakvih vizuelnih modela koji pokazuju povezanost i odnos između različitih objekata poznato je kao teorija grafova. U najobičnijem grafu, krugovi predstavljaju objekte poznate kao čvorovi, koji su povezani linijama koje se zovu grane. Istraživanja koja se bave ovakvim grafovima pitaju se: „Koliko je ovaj čvor udaljen od onog?” „Koliko grana imaju najpopularniji čvorovi?” „Da li postoji veza između ova dva čvora, i ako postoji, koliko je dugačka?” Slični grafovi se koriste da se označe mape komunikacionih mreža, no one se mogu koristiti da označe bilo kakvu mrežu, od povezanosti puteva u gradu i društvenih veza među ljudima, do hemijskih reakcija među proteinima ili širenja epidemije kroz različite lokacije. Dakle, naoružani ovim tehnikama vraćate se u piramidu. Izbegavate stražare i nadzorne kamere, infiltrirate na šesti sprat gledano odgore, nalazite skrivenu tablu, povlačite par upadljivih poluga, i usmeravate zrak smrti ka okeanu. Sada, vreme je da odgonetnete zašto vam vaš nadzorni tim uvek daje šifrovane informacije. Zdravo svima. Ako vam se svidela ova zagonetka pokušajte da rešite i ove dve.