Articles of recreational mathematics

Prove that it is impossible to have integers $b=5a$ under a digit re-ordering

Let $a$ be a positive integer and let $b$ be obtained from a by moving the initial digit of $a$ to the end. Prove that it is impossible to have $b=5a$.

General and Simple Math Problem.

This question already has an answer here: Riddle (simple arithmetic problem/illusion) 1 answer

Prime factor of $14 \uparrow \uparrow 3 + 15\uparrow \uparrow 3$ wanted

Checking the prime factors of the numbers $$z(a,b) \ := a \uparrow \uparrow 3 \ +\ b \uparrow \uparrow 3 \ ,$$ a,b positive integers with a < b the number $$z(14,15) = 14 \uparrow \uparrow 3 \ +\ 15 \uparrow \uparrow 3 = 14^{14^{14}}+15^{15^{15}}$$ turned out to be a candidate surviving trial division up […]

Super palindromes

Can anybody be kind enough to explain what exactly is a super palindrome? Also consider the following example : $923456781-123456789=799999992:9=88888888$ The largest prime factor of $88888888$ is $137$ $88888888:137=648824:101$ ($101$ being largest prime factor of $648824$) = $6424:73$ ($73$ being largest prime factor of $6424$) this in turn equals to $88$ (a palindrome) Finally, $88:11$ […]

Finding Grandma's Car – a word problem

This word problem came up in a lunchtime discussion with coworkers. None of us are professional mathematicians or teachers of math, and we weren’t sure how to get the answer. The word problem goes like this: Grandma Q drove her car downtown to do some shopping and parked it at a random spot on the […]

Convergence of $x_n = \cos (x_{n-1})$

I define the sequence $x_n = \cos (x_{n-1}), \forall n > 0$. For which starting value of $x_0 \in \mathbb{R}$ does the sequence converge?

Why does the strategy-stealing argument for tic-tac-toe work?

On the Wikipedia page for strategy-stealing arguments, there is an example of such an argument applied to tic-tac-toe: A strategy-stealing argument for tic-tac-toe goes like this: suppose that the second player has a guaranteed winning strategy, which we will call S. We can convert S into a winning strategy for the first player. The first […]

Find the largest number having this property.

The $13$-digit number $1200549600848$ has the property that for any $1 \le n \le 13$, the number formed by the first $n$ digits of $1200549600848$ is divisible by $n$ (e.g. 1|2, 2|12, 3|120, 4|1200, 5|12005, …, 13|1200549600848 using divisor notation). Question 1: Find the largest computed number having this property. Question 2: Is there a […]

About $142857$: proof that $\;3 \mid 1^n + 4^n + 2^n + 8^n + 5^n + 7^n$

Most of you already know the flabbergasting properties of cyclic numbers. Among these, $142857$ deserves to be the most famous and admired, being (in base $10$) the smallest and the only one without those spoilsport leading zeros. I personally love a couple of elegant features of its digits that aren’t often underlined. Their sequence defines […]

Help explain a new theory on small sines

(10 Mantissa[sin(10^(-100 – r1/x))])^(r2x) The reason for the argument form .10^[-n-(1/x)] is the beautiful pattern found in sin(10^-n) for positive integer n. $$ \begin{array}{| c | r |} \hline n& sin(10^{-n}) \\ \hline \\ \hline 1& 9.98334166\cdot10^{-2}\\ \hline 2& 9.99983333416666468\cdot10^{-3}\\ \hline 3& 9.9999983333334166666646825\cdot10^{-4}\\ \hline 4& 9.9999999833333333416666666646825396\cdot10^{-5}\\ \hline 5& 9.99999999983333333333416666666666468253968254\cdot10^{-6}\\ \hline 6& 9.999999999998333333333333416666666666664682539682539100\cdot10^{-7}\\ \hline 7& 9.9999999999999833333333333333416666666666666646825396825396828152\cdot10^{-8}\\ […]