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Take the subset of natural numbers that are the triangle numbers.
Find the least member of the triangle numbers such that the number
of divisors is greater than 500. The number of divisors includes
1 and the number itself.

Note that a triangle number is of the form n(n+1)/2, with n being
a member of the natural numbers.

Another way of computing the sequence of triangle numbers is:
1: 1
2: 1 + 2
3: 1 + 2 + 3
4: 1 + 2 + 3 + 4
5: 1 + 2 + 3 + 4 + 5
etc.

The basis of this problem is a square 5 by 5 grid. This represents a
map of Duonia and the title of the problem has to do with having a pilgrim
traverse the grid from the upper left hand corner to the lower right
corner. Traversing the grid has an attendant cost calculated as follows:
Traveling right (east) adds a cost of 2 units.
Traveling left (west) subtracts a cost of 2 units.
Traveling down (south) multiplies the current cost by 2.
Traveling up (north) divides the current cost by 2.
Once a segment is traversed between adjacent grid points, it may not be
taken again.
The adjective in the title has do with finding a path such that the total
cost is zero. The problem starts with the fact the penniless pilgrim
has taken two grid steps to the east and so carries a cost of 4.0 so far...

List all the perfect totient numbers less than 10,000.  A totient number is
the iterated sum of the number's totients.  A perfect totient number is a
totient number that is equal to the number itself.
But what are totients?  The totient of N is defined to be numbers less than
N and also coprime to N (note, 1 is coprime to all N).
Finally, to get back to what is meant by "iterated sum of the number's totients":
Letting the number be 327 and T be the totient function, we have that sum being
T(327) + T(216) + T(72) + T(24) + T(8) + T(4) + T(2) = 216 + 72 + 24 + 8 + 4 +
2 + 1 = 327 (which means 327 is a perfect totient number).  Note that the iterated
sum for the number 327 is derived from T(327) = 216, T(216) = 72, T(72) = 24, etc.

A k-almost-prime is a number > 1 that is product of k primes.  A squarefree
k-almost-prime is a number that is the product of k distinct primes.
Find the set of the first 10 k-almost-prime numbers, then the set of the first
squarefree k-almost-prime numbers, where k is taken from the range [1, 5].

A Collatz Sequence is one that starts with a given natural number, N.  It then
proceeds by N/2 if N is even, 3N+1 if N is odd.  The Collatz Sequence terminates
when reaches the value 1.  The Collatz Conjecture is that all Collatz Sequences
terminate.
Find the first Collatz Sequence that contains 65 or more primes.

This is to do with the Collatz Sequence.  Its tendency to go up-and-down repeatedly
like hailstones in a thundercloud suggests the title for this article.
For this item, selected the task of finding the number with the maximum sized
Collatz Sequence for the range 1..1,600,000.

Take a fraction n/d with n > d > 1 and let c be the ceiling of the fraction (the
ceiling is the smallest integer greater than or equal to the fraction).  If d does
not divide n evenly, then multiply the fraction by c giving a new value for n.
Repeat the process of the last sentence until d does divide n evenly.
The map f(n/d) = n/d * c where c is the ceiling of n/d, is referred to as the
approximate squaring map for obvious reasons.  It is conjectured but not proven
that the iterative mapping process will always terminate (i.e., d will eventually
divide n evenly).
As reported in the original Programming Praxis article, the number of iterations
is chaotic:  8/7 terminates in 3 steps resulting in a final value of 48,
6/5 terminates in 18 steps with a final value that is 57735 digits long, and
200/199 goes to a number with 10^435 digits (I think I will pass on this one...).
Implements a naive algorithm for now, no attempt at reducing due to common factors
and other possible improvements.  Uses Boost multiprecision library.  Using Boost
1.66.0, it doesn't seem to compile with gcc option -std=c++20, but it does work with
-std=c++17.

I have an easier time understanding the following definition of Motzkin numbers:
For a given natural number N, the number of distinct grid paths leading from
point (0,0) to (N,0) using as steps (1,-1), (1,0), or (1,1) with the proviso that
the path never wander into negative territory.  For example, starting at (0,0) we
add (1,1) end up at (1,1); if we add (1,1) again, we arrive at (2,2).  If instead
we add (1,-1) to (0,0) we get (1,-1) - this path is not allowed (see above proviso).

Find a 10-digit number, with all digits unique, such that the first n digits of the
number are divisible by n.  That is the first digit should be divisble by one, the
second two digits form a two-digit number that is divisible by two, the number formed
by the first three digits is divisible by three, etc.  It turns out there in only one
number that meets all the critera for a 10-digit number.

Given four points with integral coordinates determine if they form a square.
The degenerate square that consists of four repetitions of a single point may be
considered either square, or not.

Two 3-digit numbers are added together to give a 4-digit value (10 digits total for
the expression).  What values are possible using different digits (0 through 9) across
the 10 digits of the expression (i.e., each digit appears once, but no more than once)?
The leading digit of the 3-digit and 4-digit numbers is assumed not to be zero.