We are experiencing the most glorious Golden Age of mathematics ever in history.
And yet, mathematics is also at a low.
Pascal’s Triangle arises in a very natural way when we expand the powers of $x + 1$. In this short article, I want to show you just a small sample of the huge number of remarkable patterns that can be found in this triangle of numbers.
Suppose that a person wants to map a cubic equation in x so that a given one of its roots (i.e. solutions) now lies in the origin ($x = 0$). Which mapping function is best suited for this task? Suppose that this person changes their mind and now wants to place the root at $x = 1$ for instance.
Generic rules for divisibility by small integers in the decimal system are well known and commonly used due to their simplicity. For instance, a number is divisible by 2 (i.e.
Solutions by Denis Potapov, UNSW Australia.
Q1501 Find the sum of the sum of the sum of the digits for the number $2016^{2016}$.
Q1491 Find the 400th digit after the decimal point in the expansion of $(\sqrt{20} + \sqrt{15})^{2016}$.
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