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For the more visually inclined, arrange all positive integers in a 5-wide chart, as follows: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 ....


19

I found the book: M.Gardner "Aha. Insight.", p. 153. Here is the crossword, which was called there "World's smallest crossword": The clues are: Horizontal: 1. Insect 4. To annoy 5. Eavesdropper Vertical: 1. Stingers 2. To employ 3. Gigi has two (There are typos in numbers 4 and 5). Solution:


19

Shkeil got all of the nontrivial answers, but he forgot the simplest, so I'll add it here: For completeness, I'll add the rest of my thought process.


19

I think there might be quite a lot of possible answers, but here is one of them: EDIT: I think there might be quite a lot of possible answers, but here is one of them Let's rephrase that to: "I think there are barely any possible answers, but here is the easiest one." I tried to come up with another one, but the only one I was able to find eventually ...


17

There is no paradox. The teacher will get paid, one way or another. The key to understanding the situation is realizing there are multiple slightly different scenarios that are all being described as being identical: The student is obligated to pay the teacher. The student will be obligated to pay the teacher. The conditions of the contract have been met. ...


16

The problem in modern English might be stated as follows: A farmer wanted to buy three hundred oxen. He bought them at a price of $3$ oxen for $\$63$. Afterwards, he sold the oxen at $3$ for $\$63$ as well, but managed to make a profit of $\$787.50$. How did he manage this? At first blush, as you said. it appears that the farmer should have ...


16

The largest solution you cannot obtain is: Reason: Additionally:


16

Legend: 1..9 are the trees. X = empty space. Solution 1: Lines: picture: Solution 2 (built with my awesome ms paint skills): Lines:


15

The best answer I can think of in a serious vein is the Sphinx's riddle, solved by Oedipus in ancient Greece way back in the mists of time. This is surely the oldest riddle if not the oldest puzzle. Here's the story from Apollodorus 3.5.8: Laius was buried by Damasistratus, king of Plataea, and Creon, son of Menoeceus, succeeded to the kingdom. In his ...


14

What a nice question! Of course, there is no perfectly correct answer, as our historical knowledge is far too incomplete. I list some puzzles from before 1000 BC. 2700 BC: Carved stone balls. They show all regular polyhedra and the cubo-octahedron. There purpose is unclear, but they are usually listed as the starting point of recreational mathematics. ...


14

There is Reasonning.


13

(I'm assuming throughout that the concentration of wine at the end needs to be 50%.) Unless I'm missing something: There are $n$ pints of wine to begin with. After the first 3 pints are taken, you have an $(n-3)/n$ fraction of wine after the water is put back. After the second 3 pints are taken and replaced, you just multiply by the same fraction again, so ...


12

I've completely rewritten this post to hopefully be more coherent. The geometric puzzle suggests a related combinatorial puzzle: Given nine objects, how many non-isomorphic collections are there of ten sets of three objects each, such that no two of the sets have more than one object in common? Where two collections are isomorphic if there is a ...


12

Well, I'm a bit confused but I'd say With the following reasoning


11

It's not hard to construct a 1x1 crossword: +---+ |1 | | | +---+ Across: A Roman one. Down: The shortest pronoun. There's also a cheating one going around, with clues "What letter am I thinking of?" and "What is the answer to 1 Across?", and claimed solution "I'm thinking of U".


11

Case 1. Case 2. Bonus: [EDIT] To answer the question about who's right and wrong...


11

There is an article by one Pierre Ageron entitled "Le partage des dix-sept chameaux et autres exploits arithmétiques attribués à l'imâm ˁAlî. Mouvance et circulation de récits de la tradition musulmane chiite" which, if my French isn't too dodgy, means something like "The division of the seventeen camels and other arithmetical exploits attributed to the imam ...


10

The twist is that: Now the sequence of white moves is: It can be interrupted by In that case white needs one extra move:


9

Following up on the suggestion by Dennis Meng, I'm posting my suggestion as an answer instead of as part of the question. So here are my thoughts on the problem. (1) As Joe Z comments, the cattle owner needs to increase prices by one eighth in order to make the stated profit. (2) For some reason, the price is given as 63 daler for 3 oxen, rather than the ...


9

It has been dated back as far as 1933, published in a book, Diversion and Pastimes by R.M. Abraham. Source: http://www.psychologytoday.com/blog/brain-workout/200912/where-is-the-missing-dollar


9

The general picture here is as follows: if you have two positive integers $m,n$ with no common factor then every integer bigger than $mn-m-n$ can be written as $am+bn$ with $a,b$ non-negative integers, but $mn-m-n$ itself can't. Proof: First, suppose $mn-m-n = am+bn$. Then $(b+1)n$ is a multiple of $m$, and therefore (since $m,n$ have no common factor) so ...


8

Lights Out puzzles have a polynomial time algorithm by linear algebra. "Draw this shape without picking up your pencil" puzzles have a polytime algorithm which follows from the proof of which graphs have Eulerian circuits. Sliding puzzles like the 15-puzzle are easy (though hard to solve in a minimal number of moves). IMO these examples kind of prove your ...


8

I suspect the complexity class is not too important. In the example of Sudoku, the NP-complete problem is "given a partially filled in $n \times n$ grid, state whether it can be completed to a legal arrangement". The popular puzzle is "given a partially filled in $9 \times 9 $ grid that is known to have a unique solution, find that solution". In the ...


8

Napoleon in George Orwell's Animal Farm: Four legs originally (naturally). Then starts walking on two. Then is propped up by the blood, sweat and tears (three things you see) of the proletariat beneath him. Further explanation for those not familiar with Animal Farm: Napoleon was the pig in charge of a revolt that led to the animals running their own farm....


8

First things first: we will declare some constants: $\text{COW_GRASS_PER_WEEK}=$ quantity of grass eaten by a cow in one week $\text{ACRE_GRASS_PER_WEEK}=$ quantity of grass that grows in one acre in one week $\text{INITIAL_GRASS_PER_ACRE}=$ initial quantity of grass per acre We will call them respectively $CGW$, $AGW$ and $IGA$. The variables are going ...


8

Let $x$ be the starting amount of pints of wine. Now just focus on the amount of water in the barrel. After the first step there is 3 pints of water in the barrel. Because the content of the barrel is still $x$ pints the amount of water per pint is therefore $3/x$ So in the second step we take $3$ times $3/x$ water so we take away $9/x$ water and we add $...


8

Same answer as Shkeil shown in a different way.


8

Edward and Bruce After spending quite a while researching Robert the Bruce, his younger brother Edward Bruce, and other related historical figures, I hit upon the idea of considering Turning my back on you A la a dramatic insult Like Edward and Bruce QUWZTF Z'QAQ UXTRO SXURM YYYQZ MMMD' Clearly, the word to finish off I spoke for him But I am not big ...


7

In early mathematical puzzles, the "inventory problem" is addressed in the Egyptian Rhind papyrus, and may go back to 2650 BC / BCE.


7

The answer is $14.542$ pints if he replaces wine with water after taking 3 pints of wine. Let say you have 300 units of wine and you take off 3 pints of wine and add 3 pints of wine 3 times. I. Theft After you take off 3 pints of wine (3p) and add 3 pints of wine and add water the new concentration of wine would be; $\frac{300-3p}{300}$ or $\frac{100-...


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