# Tag Info

Accepted

### Can you find a 3x3 white square somewhere in this relatively prime graph?

One solution I've found Construction
• 138k
Accepted

### Which expression is larger?

EDIT: Even simpler and stronger argument:
• 11.8k

### An Amazing Configuration

I think the answer is The solution is given by the image below: This was found by @quarague asked about the case where some numbers can be equal but not all of them. In this case, the answer is
• 16.7k
Accepted

### Self-referential sequence that is sometimes powers of two

The sequence is defined as: For example, This explains the note that we the sequence can't start with two of the same number because To answer the questions: What comes after $115$ in the second ...
• 27.4k
Accepted

I believe the largest gap is Here is such an example (there may be an earlier example of a similar sized gap): To see why this is the biggest such gap,
• 9,654

### Melissa's Numbers

Update: I extended my calculation to $10^9$. I didn't find any longer runs than before, but I found another run which matched my record: Going much further will require something smarter than a brute ...
• 7,260
Accepted

### 75 integers are squared or cubed: minimum distinct results?

The minimum quantity of different results is This is because To concretely achieve this minimum, we can
• 7,260
Accepted

• 3,456
Accepted

### 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 ...
• 27.4k

• 438
Accepted

### magic square operations

A very neat puzzle! Before we dive in, a couple of lemmas/corollaries: Armed with Lemma 1 we get: Applying our corollary to column 3 forces Now This forces This is where we bring in lemma 2 Letâ€™...
• 1,197

### Geometry Puzzle: Tangent Circles with Integer Radii

This puzzle has already been solved using 30-digit numbers, but I'm interested in using smaller numbers. I wrote some code to start with a central unit circle and the loop over values for the next-...
• 1,738

### sums and differences in consecutive grid

Very nice puzzle! @WeatherVane beat me to the answer but thought I'd post with some logical deductions: Logic on how to solve:
• 61.9k

### The Bogotá Marathon

Some comments on part (b) (Warning: highbrow mathematics ahead, done by someone not actually particularly expert in the relevant field. Non-mathematicians will be intimidated, actual experts will ...
• 121k

### Which expression is larger?

The larger expression is Proof: Remark:
• 2,135

### Primeable numbers

Allowing for zeros to be discarded to form primes with fewer digits, the first sequence containing ten or more consecutive non-primeable numbers is: Related sequences: Searching all numbers with six ...
• 16.2k

### magic square operations

My solution, also showing the sums Found by recursive algorithm, before another answer was corrected. The solution is unique.
• 14.7k

My solution:
• 14.7k

### Melissa's Numbers

I confirm @isaacg's results. I found some more sequences of length 6 that may be useful for spotting a pattern: Also I believe there are infinitely many quadruples!
• 36.5k

### While 2024 arrives

I'm not convinced this is optimal, but I think this satisfies part i: Why is the partition unique?
• 3,456

### Can you find a 3x3 white square somewhere in this relatively prime graph?

Cool puzzle! In my notation $(N,M)$ is the top-left corner of the white sub-grid. I used a computer program to find 99 solutions for $N \leq M \leq 10000$: I couldn't find any 4x4 or 14x1 solutions ...
• 36.5k

### Primeable numbers

Let's try to understand the probability of very large unprimeable numbers. To do so, let's start by looking at how many permutations a number has. If a number has only a single unique digit, it has ...
• 7,260
Accepted

### Intermingled primes

The 6 different primes are: Deductions (I solved it with regular expressions in a list of all prime numbers from 101-997):
• 8,357

### Vector Sum of Pythagorean Triples

By definition of (Euclidean) length, 2-dimensional vector $v_j:=(a_j,b_j)$ has length $|v_j|=c_j$ if and only if $a_j^2+b_j^2=c_j^2$ ($c_j\ge 0$). By repeated application of the we have for any ...
• 21.4k
Accepted

### mathematical magic trick

The short version of how/why it works:
• 8,590
Accepted

### Number operation problem - find minimum number of operations

I can do it in seven operations: Alternative path, with smaller final numbers: EDIT: A quick computer search revealed what I should have seen by hand:
• 11.8k

### Geometry Puzzle: Tangent Circles with Integer Radii

edwardh's answer above gives eight circles with radii circa $10^{42}$: ...
• 2,188
Accepted

### First five digits of a googol factorial

I think the answer is So, Stirling's formula says that $n!\simeq\frac1{\sqrt{2\pi n}}(n/e)^n$, so the answer we're looking for almost certainly agrees with the first five digits of the RHS. The \$1/\...
• 121k

Only top scored, non community-wiki answers of a minimum length are eligible