# Questions tagged [recreational-mathematics]

Mathematics done just for fun, often disjoint from typical school mathematics curriculum. Also see the [puzzle] and [contest-math] tags.

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### If we are drawing red and black cards out of an infinite deck, and we draw red with probability 4/5, what is E(num draws to draw 3 consecutive blacks)

This is the full question: John is drawing red and black cards out of an infinite deck. The probability of drawing a red card is 4/5. Calculate the expected number of draws to draw three black cards ...
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You have 8 identical balls and one is slightly heavier. Using a balance whats the minimum number of times you need to weigh yo find the heavier ball
1 vote
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### Prove $\sqrt{\frac{a+6bc}{a+6}}+\sqrt{\frac{b+6ac}{b+6}}+\sqrt{\frac{c+6ab}{c+6}}\ge 3.$

I found the inequality here (#25) : Let $a,b,c$ be nonnegative real numbers such that $ab+bc+ca+abc=4$ Prove that $$\sqrt{\frac{a+6bc}{a+6}}+\sqrt{\frac{b+6ac}{b+6}}+\sqrt{\frac{c+6ab}{c+6}}\ge 3$$ I ...
125 views

### Conjecture: The sum of the coefficients of terms in the $n$-th derivative of $\sec (x)$ is $n!$ (and another pattern)

Let $f^{(n)}$ stands for the $n$th derivative. The conjectured pattern is: The summation of coefficients of the terms of $(\sec {x})^{(n)}$ equals $n!$ For example: $(\sec {x})^{(0)}=\sec (x)$, the ...
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### Chessboard Domination by too few queens?

I am looking for patterns of Domination by Independent queens - one source says there is a pattern for 7x7 with only 4 queens with no queens attacking another ie independent. I cannot find it so far. ...
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### Solutions to $(f(x)-f(y))^3=f\left(x^3\right)-f\left(y^3\right)$

I was wondering, if there are more solutions to the functional equations, than $f(x) = const$. Maybe someone has an idea of how to find all solutions (or all continuous solutions)? Find all the ...
27 views

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### Prove that the numbers 2008 and 2106 are not terms of this sequence.

The sequence $(x_n)$ is given recursively with $x_1 = 188$, and $x_{n+1}$ is obtained from $x_n$ by adding twice the sum of the digits of the number $x_n$. Prove that the numbers 2008 and 2106 are not ...
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### Logic Puzzle with truths & lies to questions

Puzzle : We are at a crossroad of 2 paths. One path leads to a swamp and one path leads to the treasure. At the crossroad we meet 2 people , of whom we know that 1 is always telling the truth and 1 is ...
161 views

### Counterexample for a proof

Let $n$ and $k$ be positive integers and $$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$ be a lattice cube of length $n$. Suppose that $3n^2 - 3n + 1 + k$ points of $T$ are colored red ...
100 views

### Example of functions $f$ and $g$ where $f\circ g(x) = x^2$ and $g \circ f(x) = x^3$ for range $(1, \infty)$ [closed]

I'm just looking for examples of real functions $f$ and $g$ where $f \circ g(x) = x^2$ and $g \circ f(x) = x^3$ and the domains and codomains of $f$ and $g$ is $(1, \infty)$. No complex functions but ...
1 vote
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### Determine the minimal tiling, allowing for both overhang and overlap, from a small shape to a larger one.

Context of Concrete Problem: While I have run into similar problems in other games, this specific one is for the game Stardew Valley. You would like to make a farm involving a scarecrow and the lowest ...
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