Find the sum of all the multiples of 3 or 5 below 1000.

Let us generalize again to a finite set of factors.

There is a formula for the power of finite sets

Here is a measure (i.e. it commutes with the union sign) and can be replaced with --- power of a set sign or, if we are in the natural numbers space, with the sum of elements sign, as in our case. is not a multiindex, but a subset of the natural numbers cut from to .

Now by we denote the set of all the multiples of factors , less than certain number N, where i varies from 1 to n (each is respectively the set of multiples of a factor ). We use the above-mentioned formula to compute the measure of the union via measures of all and measures of all finite intersections of them.

Suppose, we have a number , prime or not, and the set of all it's multiples (they include only numbers less than N). Power of this set is of course (div operation) and the sum of its members can be calculated by the well-known formula for the sum of an arithmetic progression.

As regards all the intersections, it is understandable that we ought to calculate the least common multiple (LCM) of taken factors, and the set-intersection of their multiples will be just a set of its multiples. However, the current version of the solution assumes that we take primes as factors, then the LCM of them is just their product. When I calculate proper LCM in Problem 5 (up to now there is a bruteforce version), I will switch the temp version to it.

Let's see the solution. New util functions:

The function that calculates subsets of a set:

Important thing about this function is that it returns the empty set as the first element and the full set as the last element of a result list, all other subsets are in between. The number of subsets of a finite subset is just , so the complexity is --- it would be better visible with an imperative-iterative version of this function (I'm not posting it here). As regards memory, the function generates all the subsets as lists which in whole contain elements (strange, this neat formula isn't on Wikipedia yet, I should add it there), that is the memory load is . This is a not-so-good idea to load everything into memory, as we can rewrite this function (and the function that is down here in the post) iteratively with O(n) memory complexity --- taking advantage of combinadics, but for now I am satisfied enough with this version.

Using the formula above the solution now as simple as

With that (cdr) I cut off the empty subset, whose measure is zero (otherwise the (sum-of-one) function has to be a bit more complex).

Let's be careful with notation: n here is actually not the same n, as in the (subsets) function, but the number up there, the maximum of our multiples-sets. The performance complexity depends on k and N, but we are interested only in complexity, depending on N. Let's assume that k is small comparing to N, which should be the usual case. Then the complexity is roughly speaking O(1), doesn't depend on N, as we wanted (I remind that in the previous version we had O(N) complexity).

The final touches are the regression tests:

The last commented one breaks, of course, as 15 is not prime - the LCM algo should be updated still.

I've looked through the Quick: An Introduction to PLT Scheme with Pictures document. And what I've learned is:

• There is a library (#lang slideshow) embedded into PLT-Scheme, which provides some easy-to-use graphic primitives and a GUI library (scheme/gui/base). The first can be used for drawing on GUI's canvases.
• There is an OOP library (scheme/class). I should look what's the backbone later.
• PLT has the distribution system for libraries. The first eval of (require (planet something)) downloads from the PLaneT server and caches 'something' locally.

The last is quite nice, I should run through that server and look which libraries are actually implemented.

Find the 10001st prime.

The problem of finding primes is the most famous computational task in the number theory.

At first, let's update the (generate-list-iteratively) function. It now includes i - the parameter, which denotes number of eligible elements up to this moment. We use it in the stopping condition.

Dependant functions are rewritten also. I refactored the factorization functions and added primality tests (usual straightforward ones):

Now, having an extra function that returns the last element in a list, we can easily solve the 7-th problem:

The complexity in memory is O(n), as we store n primes. It could be O(1), of course, but I prefer this way for now. The performance complexity of the algorithm is hard to determine. Probably, I need to overlook some literature for that.

What is the difference between the sum of the squares and the square of the sums?

No problems with brute-force here.

The number of operations (additions and multiplications) here is 3*n + 2. The complexity is O(n), and the algo is O(n) in memory as well (of course, can be done with O(1) in memory).

What is the smallest number divisible by each of the numbers 1 to 20?

This is also quite known problem --- finding a least common multiple (LCM).

As usual, let's start with the brute-force solution.

This is the most inefficient way we can invent for the task of finding LCM. The complexity in the worst case is something like O(p^n), when n divisors out there are prime numbers, each is p at the average (but different, of course, then their LCM is just their product). More strictly, the number of divisions is up to n*p^n in the worst case (would be that if not shorcutted logical operations in (all-divide?) function).

Find the largest palindrome made from the product of two 3-digit numbers.

An additional function that filters out empty lists from a list of lists:

The function that represents a number in arbitrary number system and the function that returns 10-base digits for a number (starting from the rightmost digit):

(digits) function could use standard (number->string), but this generalized solution I like more. Now a solution to the problem looks as easy as:

Notice, for comparing lists we use (equal?), not (eq?) and not (eqv?) - they would return just #f. However, we could use (eq?) instead of (null?) in (filter-empty-lists) to compare a list to the empty one. As regards number of operations, we have n^2 / 2 multiplications, as many (palindrome-number?) calls that create digits lists (up to 6 divmod per each) and compare them (up to 6 comparisons per each). So, the complexity is O(n^2), and memory use is proportional to the palindrome numbers density in the n by n matrix (half of which we actually iterate over to find an answer).

Problem 3

Find the largest prime factor of a composite number.

is a very well known problem called factorization. The fundamental theorem of arithmetics comes in handy.

Brute-force in this case is just iterating from 2 upwards, searching for factors:

If a factor is found, we divide the number by it, and continue with searching factors of a quotient. We don't need to start from 2 again (see the theorem).

The complexity for finding one factor is O(n) in the worst case. Let us optimize at once, stopping the iteration at sqrt(n). Indeed, a number cannot have a factor greater than its square root. This improves the worst case complexity drastically to O(\sqrt(n)) and this is the classical factorization algo.

Now, having written additional function (max-list) for finding maximums through lists, we can solve the third problem:

I've written the generic procedure to generate lists iteratively:

I've refactored (create-filtered-numbers-list from to filter) function into (numbers-filters) and added also (numbers) for future use:

Now the brute-force solution for the problem 2

Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed four million

looks like this:

The complexity is O(n). I decided not to generate a filtered list, but rather first generate, then filter, (fibonacci) function goes to the library for future use (for future functions I will not mention placing into my library). This is an optimal algorithm for generating a Fibonacci sequence, unlike computing the n-th member of it, which can be done more efficiently. However, for computing a sum of members, perhaps, there is a better solution, I will think about it later.

I refactored the previous solution and extracted such library functions, they could be useful in future:

I think for these algoritms, where we have to iterate through large lists of numbers, the laziness (e.g. in Haskell) would give slightly more elegant code at certain places.

The first problem solution now becomes:

For a list of divisors with length k the number of operations will be k*n. Actually, less than that because of shortcut (exists). Assuming k fixed and small, the complexity is still O(n). We could make use of the not yet written (create-numbers-list) function and filter that afterwards instead of the (create-filtered-numbers-list) function, but let us take less space in memory at once.

It took me quite a while to understand. Today I'm using a laptop with Windows, so here is solution for the windows-version of the PLT Scheme implementation. Under Linux it will not differ much. By the way, to use R6RS in PLT DrScheme you need to choose Module language (Ctrl+L) and write #!r6rs in the beginning of each program.

First, write your own library. I separated the function (divides?) from the previous program:

Second, save it under the path, for example, C:\Projects\projecteulersolutions\project-euler-lib.scm and, as it is explained in "C:\Program Files\PLT\doc\r6rs\Installing_Libraries.html", compile and install the library into collections folder by running the command (the first way explained there seems not to be working):

"C:\Program Files\PLT\plt-r6rs.exe" --install C:\Projects\projecteulersolutions\project-euler-lib.scm

Some files drop into the directory "C:\Users\User\AppData\Roaming\PLT Scheme\4.1.4\collects\project-euler-lib"

Third, you connect it to the program by replacing (import (rnrs)) clause with (import (rnrs) (project-euler-lib)). And, of course, delete (divides?) function form the top-level program body.

Fourth, before recompiling a library, don't forget to clean the folder:

rmdir /s /q "C:\Users\User\AppData\Roaming\PLT Scheme\4.1.4\collects\project-euler-lib"