New answers tagged mathematics
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Irregularly Deposited Compound Interest
Consider an interval, over which a newly-deposited balance of $b$ accrues an interest amount $i$.
Why?
Let's crunch some numbers!
And now, a simple program: invest(balance, interest rate, time) ...
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Irregularly Deposited Compound Interest
Observation: Let's for the moment assume we know the optimal number of transfers and need only optimise the timing. Freezing all but one transfer (#k, say) we find that its best timing $t_k$ only ...
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Nimber mnemonic combinatorial puzzle
There are 384 solutions. Here's one:
I used integer linear programming as follows. Let $$P=\{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o\}$$ be the set of positions, where position $o$ must take ...
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Irregularly Deposited Compound Interest
Not sure if I'm right here, but this is my best solution.
First of all, in my solution:
So, with that, I came up with the following formula:
Since I wasn't able to think of a way to expand that ...
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Rearrange words to make a sentence
no triangle has more than two angles which are not less than seventy degrees
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Mishustin's circle problem
A fairly simple solution is to place the compass on the point, and draw a circular arc (of arbitrary diameter) which intersects the diameter (possibly extended, or "produced", to use the ...
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Mishustin's circle problem
Here's my go (click to embiggen)
Steps:
Connect A to P and pick an arbitrary point Q between them, near-ish to P.
Then, draw lines as shown, constructing the points in alphabetical order, which ...
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Mishustin's circle problem
Note: This is not a valid answer, given the poster's clarification on what the straightedge is capable of. I'm leaving it up because I think it's interesting.
I found a way to solve this, based on
...
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Nimber Mnemonics
Multiplication of nimbers between $1$ and $15$ (or between $0$ and $2^{2^n}-1$ for any $n$) has a primitive root: a number whose powers generate all the nimbers we want. (In fact, I believe that $2^{2^...
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Pursuit-evasion game
First of all, how can the criminal even get caught?
Can the cops arrange this scenario?
How long does this take?
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Create a permutation with longest increasing subsequence length 3
Another solution, which you might perhaps regard as not so simple:
10, 20, 30, 9, 19, 29, ..., 1, 11, 21.
An increasing sequence $a, b, c$ needs $a<11$, $10<b<21$ and $c>20$.
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Longest subsequences and shortest longest ones
A best possible sequence is
which has a longest increasing subsequence of length 3 and a longest decreasing of length...
Another example is
which has
These are best possible because
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Create a permutation with longest increasing subsequence length 3
The following seems to be a simple solution:
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Equality-breaking function
As mentioned in juathalf's answer, the definition of the "function" $f$ is that
But what is the most inclusive possible domain of $f$? A first attempt at answering that is to simply say ...
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'SILVER' -> ‘LESIRU' and 'GOLDEN' -> 'LEGOND', so 'NATURE' -> what?
Looking at the options B and C can be easily ruled out. Either A or D is correct.
S I L V E R -----> L E S I R V (123456 ---> 351264)
Similarly
NATURE -----> TRNAEU
So done! Its option (D).
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Find the wrong number in the given series
I think the "wrong" number in the sequence is
Reasoning
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'SILVER' -> ‘LESIRU' and 'GOLDEN' -> 'LEGOND', so 'NATURE' -> what?
Breaking up the encryption into steps I get:
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'SILVER' -> ‘LESIRU' and 'GOLDEN' -> 'LEGOND', so 'NATURE' -> what?
There is a typo in the letters 'LESIRU', it should be 'LESIRV'
The trick is to give each letter a number, and then cross reference it to the original:
...
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'SILVER' -> ‘LESIRU' and 'GOLDEN' -> 'LEGOND', so 'NATURE' -> what?
SILVER -> LESIRU
GOLDEN -> LEGOND
Now do the same for NATURE and use the second sequence of numbers to rearrange the resulting word:
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'SILVER' -> ‘LESIRU' and 'GOLDEN' -> 'LEGOND', so 'NATURE' -> what?
It looks like there's a typo in the question:
If so, here is how I solved it:
Then the answer is:
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Minimum number of turns
Oo, I've got this one; these come up a lot while organising board game tournaments.
a) what is the minimum number of turns needed to determine the heaviest box?
b) what is the min number of turns ...
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What is the probability that your life will have lasted for 100 years once you die?
I feel that 10% is the easier answer to justify, but I admit it depends entirely on what assumptions one makes. Bayesians will rightly point out that it depends on how you do the "sampling", ...
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In a certain code 'NATIONAL' is written as 'JUBOKZMN'. How is 'ELECTION' written in that code?
Starting with 'NATIONAL'
In the same manner, 'ELECTION' becomes
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The Monty Hall loot box
Let w and p ("win" and "pool") the two limiting proportions. Because of the pity timer we have
If three goats are drawn we are guaranteed to lose. Otherwise our chances are 2:1 in ...
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Equality-breaking function
$f(x)$ is a function that:
Verification:
There is probably some more precision in the definition that I'm missing (to explain the last property), but I'm quite confident the general idea is in the ...
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Cooperative guessing game: no incorrect guesses
This generalize for any $N$ not necessarily of the form$2^k-1$.
Let $N$ be an integer such that $2^{k-1} \leq N \leq 2^k-1$, where $k=\lfloor log_2 N \rfloor+1$.
By this strategy one can guarantee ...
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Minimum function optimization puzzle #3: 3 functions
A short program in R confirms @DanielMathias answer.
You can try the code here.
The code in text: (not showing correctly because of characters '<' and '>')
...
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Minimum function optimization puzzle #3: 3 functions
The minimum number of steps is
With the sequence
My approach: work backwards from the goal.
The final step must be
Apply the inverse of h(x) repeatedly and look for values that are near a square.
...
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Counting puzzle #1: Function combinations
Computerless solution
There are
in the set S.
Let's first of all consider
Well, actually
Anyway, we aren't done yet, because
We now have
Now we need to augment our list by
We're still not quite ...
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Counting puzzle #1: Function combinations
Programmed solution:
These numbers are listed below, along with the function that produces them.
C code to identify and count the numbers:
...
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What is the probability that your life will have lasted for 100 years once you die?
90% of people live for 3 years, and 10% live for 100 years.
So, at any given moment, if a 3yr old 'dies' they will be replaced with a new 'to be 3yr old' because the stat of 90% must be constant. ...
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Is my solution to a mathematics puzzle I created the most efficient solution there is to it?
Let's see what an optimal path would look like. For starters,
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What is the probability that your life will have lasted for 100 years once you die?
Assumptions:
New assumption based on comment below
Answer:
On a side-note:
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What is the probability that your life will have lasted for 100 years once you die?
I will attempt a conversion to the sleeping beauty paradox.
Current post -> sleeping beauty:
...
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What is the probability that your life will have lasted for 100 years once you die?
The frequentist answer to this question is
This is because
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What is the probability that your life will have lasted for 100 years once you die?
That assumption is
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