30 votes
Accepted

The numbers on blackboard

Let $$f(a,b) := \frac{\sqrt{a^2+3ab+b^2-2a-2b+4}}{ab+4}.$$ John's procedure is now to repeatedly replace the leftmost two values $a,b$ on the blackboard by the single value $f(a,b)$. I claim that John ...
Magma's user avatar
  • 5,259
26 votes

Which is larger? $\sqrt{15} - \sqrt{7} + \sqrt{5} + \sqrt{2}$ versus $5$

The answer is I imagine the line of reasoning the author wants is as follows:
hexomino's user avatar
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24 votes
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Two Cannons - A Beginner's Physics Puzzle

Alain Remillard has given the mathematician's answer. Here's the physicist's one: Step 1: Obviously, in such a universe, regardless of their speed, the cannonballs will travel in a straight line and ...
Bass's user avatar
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19 votes
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A man bought several liters of wine

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.
Scott M's user avatar
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18 votes
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Google cannot find me

Okay so I think the answer is Google cannot find me, reflected I'm lemon and lime, googling 'me' is when you fall, me and meme will answer all.
hexomino's user avatar
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16 votes
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Board with all 2020s

It is Reasoning:
WhatsUp's user avatar
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15 votes
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The adventitious 18-gon

To illustrate the puzzle I made the following image: All angles I entered can be simply calculated using the fact any n-gon has a total of $180 + (n-3) \cdot 180 °$. Now let's call the intersection ...
Ivo's user avatar
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15 votes
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Programmatically solving a math puzzle with four unknowns and four equations

First of all, let's see why your brute-forcing fails. (This is the puzzle part, the rest is plain old math.) No matter which you chose, the number at the bottom right would have to be both odd and ...
Bass's user avatar
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14 votes

A man bought several liters of wine

There is Reasonning.
Shkeil's user avatar
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13 votes
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Turning wine into water

(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 ...
astralfenix's user avatar
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13 votes
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Vector Sum of Pythagorean Triples

The graph of real-valued Pythagorean triples $x^2+y^2=z^2$ forms an infinite cone if we restrict to $z>0$: A sum of $n$ independent vectors on this cone is $n$ times their average, which lies ...
xnor's user avatar
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12 votes
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Big number puzzle

And here is the number you are probably thinking of: It works only for $ab$ where $a \le b$. I suppose that it is a mistake in the problem statement. Others have proven that as it is, the problem ...
Florian F's user avatar
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12 votes
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Can you find the number?

The number is because we can rewrite the three given conditions as follows, where $H$, $T$, and $U$ are the hundreds, tens, and units digits respectively: Adding the first two gives Also from the ...
Rand al'Thor's user avatar
11 votes
Accepted

An ugly formula

@Tony Ruth's answer provides this alternate form of the equation: Adding 2 to both sides and factoring, Then Substituting this back in, So Add 2 to both sides again: So And For every $y>1$, ...
f'''s user avatar
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11 votes

Which is larger? $\sqrt{15} - \sqrt{7} + \sqrt{5} + \sqrt{2}$ versus $5$

Here's another approach, which starts from smaller integer residuals than hexomino's solution. and Finally, bringing these lines of argument togegther, I admit it isn't a single chain of inferences....
Rosie F's user avatar
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11 votes
Accepted

Six fruit baskets

The minimum greater than 1000 is: Because:
Paul Evans's user avatar
  • 9,431
11 votes

Two Cannons - A Beginner's Physics Puzzle

They will I did it mathematically Suppose the horizontal distance between both cannons is $d$ and the up angle from right cannon is $\theta$. Then, the left cannon is at a height of $d\tan\...
Alain Remillard's user avatar
10 votes
Accepted

And the average is 51.5625

Note: 51.5625 = 825/16 To calculate the sum for a given $n$ we can do $\frac{n \cdot (n+1)}{2}$. Then we need to have that $\frac{n \cdot (n+1)}{2} - \frac{825 \cdot (n - 4)}{16}$ is a number that can ...
Ivo's user avatar
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10 votes
Accepted

What is the ratio between small and big x-gons?

Inscribe the original $n$-gon in a circle of radius 1. The apothem of the large $n$-gon is $\cos(\frac{\pi}{n})$ and the apothem of the small $n$-gon is $\sin(\frac{\pi}{2n})$. Therefore the ratio of ...
f'''s user avatar
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10 votes
Accepted

Videogame salesmen

Edit:
Jaap Scherphuis's user avatar
10 votes

SafeCracker #3 - We've Been Blocked

The doesn't belong, because
19aksh's user avatar
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9 votes
Accepted

Blackboard problem with polynomial

The smallest possible value of $n$ is Claim: We can get every non-negative integer $n\leq 2016$ on the board. Proof: By induction. We start with $n=0$ on the board. We can get $1$ using Lord of the ...
Julian Rosen's user avatar
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9 votes

Google cannot find me

Is the answer Googling it returns me and meme will answer all. Can't make the middle two lines fit though
Mohirl's user avatar
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9 votes
Accepted

The special machine

This is probably similar to msh210's idea but if we write
hexomino's user avatar
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8 votes

A man bought several liters of wine

Same answer as Shkeil shown in a different way.
hexomino's user avatar
  • 136k
8 votes
Accepted

The grazing cows of Sir Isaac Newton

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 ...
Narmer's user avatar
  • 3,567
8 votes

Turning wine into water

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 ...
Ivo's user avatar
  • 11.2k
8 votes

Board with all 2020s

A less technical solution: We can (try to) make a symmetric solution that makes all numbers equal (where the 6 variables signify how often a cell is chosen): ...
Retudin's user avatar
  • 81
7 votes

Turning wine into water

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. ...
Oray's user avatar
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7 votes
Accepted

Iterative Floors and Ceilings

To make things more general (and make certain relationships clearer), replace $2016$ with $N$. There exists some non-negative $q,r$ such that $N = qk +r$ and $r < k$ (i.e., $q$ is the quotient and $...
Paul Sinclair's user avatar

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