The post Project Euler 140 Solution appeared first on Dreamshire.
]]>Consider the infinite polynomial series A_{G}(x) = xG_{1} + x^{2}G_{2} + x^{3}G_{3} + …, where G_{k} is the kth term of the second order recurrence relation G_{k} = G_{k−1} + G_{k−2}, G_{1} = 1 and G_{2} = 4; that is, 1, 4, 5, 9, 14, 23, … .
For this problem we shall be concerned with values of x for which A_{G}(x) is a positive integer.
The corresponding values of x for the first five natural numbers are shown below.
We shall call A_{G}(x) a golden nugget if x is rational, because they become increasingly rarer; for example, the 20th golden nugget is 211345365.
Find the sum of the first thirty golden nuggets.
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]]>The post Project Euler 147 Solution appeared first on Dreamshire.
]]>In a 3×2 cross-hatched grid, a total of 37 different rectangles could be situated within that grid as indicated in the sketch.
There are 5 grids smaller than 3×2, vertical and horizontal dimensions being important, i.e. 1×1, 2×1, 3×1, 1×2 and 2×2. If each of them is cross-hatched, the following number of different rectangles could be situated within those smaller grids:
1×1: 1
2×1: 4
3×1: 8
1×2: 4
2×2: 18
Adding those to the 37 of the 3×2 grid, a total of 72 different rectangles could be situated within 3×2 and smaller grids.
How many different rectangles could be situated within 47×43 and smaller grids?
1. Count the number of horizontal and vertical rectangles in the grid.
2. Count the number of diagonal rectangles in the grid. Since m ≥ n, you can safely exchange m and n if this isn’t already the case.
3. Add the two together for a total for each iteration of m and n. The sum of this series is the answer.
Example: for a 2×3 grid you have 18 horizontal and vertical rectangles and 19 diagonal rectangles for a total of 37.
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]]>The post Project Euler 101 Solution appeared first on Dreamshire.
]]>If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next term, as there are infinitely many polynomial functions that can model the sequence.
As an example, let us consider the sequence of cube numbers. This is defined by the generating function,
u_{n} = n^{3}: 1, 8, 27, 64, 125, 216, …
Suppose we were only given the first two terms of this sequence. Working on the principle that "simple is best" we should assume a linear relationship and predict the next term to be 15 (common difference 7). Even if we were presented with the first three terms, by the same principle of simplicity, a quadratic relationship should be assumed.
We shall define OP(k, n) to be the n^{th} term of the optimum polynomial generating function for the first k terms of a sequence. It should be clear that OP(k, n) will accurately generate the terms of the sequence for n ≤ k, and potentially the first incorrect term (FIT) will be OP(k, k+1); in which case we shall call it a bad OP (BOP).
As a basis, if we were only given the first term of sequence, it would be most sensible to assume constancy; that is, for n ≥ 2, OP(1, n) = u_{1}.
Hence we obtain the following OPs for the cubic sequence:
OP(1, n) = 1 | 1, 1, 1, 1, … |
OP(2, n) = 7n−6 | 1, 8, 15, … |
OP(3, n) = 6n^{2}−11n+6 | 1, 8, 27, 58, … |
OP(4, n) = n^{3} | 1, 8, 27, 64, 125, … |
Clearly no BOPs exist for k ≥ 4.
By considering the sum of FITs generated by the BOPs (indicated in red above), we obtain 1 + 15 + 58 = 74.
Consider the following tenth degree polynomial generating function:
u_{n} = 1 − n + n^{2} − n^{3} + n^{4} − n^{5} + n^{6} − n^{7} + n^{8} − n^{9} + n^{10}
Find the sum of FITs for the BOPs.
I used the difference method to solve this one as we only need to sum the FITs and not extend the series or find equations through fitting.
Take the example and build a pyramidal array with the base of k terms of the original expression. Next, take the differences between left-to-right neighbors and stack those differences on top of the base until the triangle is complete.
Now, sum the entire array to calculate the sum of all FITs. The individual FITs are summed along the diagonal layers of the triangle:
12 + 19 + 27 = 58, 7 + 8 = 15, 1
OP = n^{3}, k = 3
12 7 19 1 8 27
12 + 7 + 19 + 1 + 8 + 27 = 74
k, u = 3, lambda x: x**3 def tld(x): if len(x[-1]) == 1: return x return x + tld([[x[-1][i] - x[-1][i-1] for i in range(1, len(x[-1]))]]) print "Sum of FITs for the BOPs", sum(sum(tld([map(u, range(1, k+1))]), []))
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]]>The post Project Euler 102 Solution appeared first on Dreamshire.
]]>Project Euler 102: Three distinct points are plotted at random on a Cartesian plane, for which -1000 ≤ x, y ≤ 1000, such that a triangle is formed.
Consider the following two triangles:
It can be verified that triangle ABC contains the origin, whereas triangle XYZ does not.
Using triangles.txt (right click and ‘Save Link/Target As…’), a 27K text file containing the co-ordinates of one thousand "random" triangles, find the number of triangles for which the interior contains the origin.
NOTE: The first two examples in the file represent the triangles in the example given above.
There are a few ways to solve this problem. With having the point to check be the origin (0,0), then the easiest way is to check that the signs of the cross products are all positive or negative. This solution could be easily modified to check points other than the origin.
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]]>The post Project Euler 114 Solution appeared first on Dreamshire.
]]>A row measuring seven units in length has red blocks with a minimum length of three units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square. There are exactly seventeen ways of doing this.
How many ways can a row measuring fifty units in length be filled?
NOTE: Although the example above does not lend itself to the possibility, in general it is permitted to mix block sizes. For example, on a row measuring eight units in length you could use red (3), black (1), and red (4).
For the purpose of this discussion, the unit size of the space into which blocks (red) are placed is identified as n and the unit size of the red blocks, with a range in size from 3 to n, is identified as m. Also, an empty space void of any red blocks is counted.
This is a counting problem, so dynamic programming and its hand maiden linear programming come immediately to mind and usurp the notion of trying a brute-force solution.
And as far as examples go, 7 wasn’t the best choice as the problem admits, but it will suffice to answer the problem.
My first analogy was shifting groups of bits in a computer’s register, but here’s a more festive idea. I looked at this problem like a game of Tetris (Play a game online here). This is a much simpler version as there is only one dropping shape, a “line” with a minimum size of 3 units. Let’s look at the example graphic twisted slightly to show my thought process:
The only constraint is that no falling shapes can ever touch; they must be separated by at least one empty space.
As the red blocks fall the only variable is the length of the block, m; between 3 and 7. When we move to n = 8, this same configuration for n = 7 will be a subset of it. Just like n = 6 is a subset of n = 7.
In fact, to demonstrate this concept, lets try to calculate the number of ways we can fill 6 spaces:
How many survived the butchery? Change the variable n in the program to 6 and run it. Did it match your count?
By now, you could add a tile to our graphic and learn the count for 8 spaces. Soon a pattern would become obvious and a dynamic solution is born.
It’s just adding a series of numbers for each n as it increase from 3 through 50. Yes, a function (with binomial tendencies) exists out there that would always put to rest a dynamic programming solution for counting, but why spoil the fun.
from Euler import binomial as C n = 50 + 1 print sum(C(n - 2*k, 2*k) for k in range(int(n/4)+1))
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]]>The post Project Euler 115 Solution appeared first on Dreamshire.
]]>NOTE: This is a more difficult version of Problem 114.
A row measuring n units in length has red blocks with a minimum length of m units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.
Let the fill-count function, F(m, n), represent the number of ways that a row can be filled.
For example, F(3, 29) = 673135 and F(3, 30) = 1089155.
That is, for m = 3, it can be seen that n = 30 is the smallest value for which the fill-count function first exceeds one million.
In the same way, for m = 10, it can be verified that F(10, 56) = 880711 and F(10, 57) = 1148904, so n = 57 is the least value for which the fill-count function first exceeds one million.
For m = 50, find the least value of n for which the fill-count function first exceeds one million.
This problem could easily be solved by playing with the interactive solution on the Problem 114 page. Just change m to 50 and play with n until it first exceeeds one million. The value of n won’t be very big because we start m at 50.
Again, i opted out of a recursive solution or generating functions because, after a while, dynamic programming becomes so intuitive for solving counting problems.
Also, I didn’t bother with an F()
function as I wanted to simply make a few changes to the previous problem and finish it. Instead of a fixed-size array, I dynamically increase it as necessary since I have no clue as to how big it should be to start with.
Well, actually, I do have a clue, but not if the parameters of this problem were changed.
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]]>The post Project Euler 116 Solution appeared first on Dreamshire.
]]>A row of five black square tiles is to have a number of its tiles replaced with coloured oblong tiles chosen from red (length two), green (length three), or blue (length four).
If red tiles are chosen there are exactly seven ways this can be done.
If green tiles are chosen there are three ways.
And if blue tiles are chosen there are two ways.
Assuming that colours cannot be mixed there are 7 + 3 + 2 = 12 ways of replacing the black tiles in a row measuring five units in length.
How many different ways can the black tiles in a row measuring fifty units in length be replaced if colours cannot be mixed and at least one coloured tile must be used?
NOTE: This is related to Problem 117.
As a continuation from the two previous problems we remove the empty space option and one-space tile separation requirement. Now it makes sense to include a function since we are adding 3 different fixed-length tiles of sizes 2, 3 and 4.
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]]>The post Project Euler 117 Solution appeared first on Dreamshire.
]]>Using a combination of black square tiles and oblong tiles chosen from: red tiles measuring two units, green tiles measuring three units, and blue tiles measuring four units, it is possible to tile a row measuring five units in length in exactly fifteen different ways.
How many ways can a row measuring fifty units in length be tiled?
NOTE: This is related to Problem 116.
As the simplest solution for the problem set requium, we make a small change to problem 116 by allowing multiple tiles of length 2, 3 and 4 to share the space.
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]]>The post Project Euler 123 Solution appeared first on Dreamshire.
]]>Project Euler 123: Let p_{n} be the nth prime: 2, 3, 5, 7, 11, …, and let r be the remainder when (p_{n}−1)^{n} + (p_{n}+1)^{n} is divided by p_{n}^{2}.
For example, when n = 3, p_{3} = 5, and 4^{3} + 6^{3} = 280 ≡ 5 mod 25.
The least value of n for which the remainder first exceeds 10^{9} is 7037.
Find the least value of n for which the remainder first exceeds 10^{10}.
Using much of our understanding from solving Problem 120, we are able to easily adapt those methods to solve this problem. The differences are as follows:
We can make an educated guess of how many primes we’ll need by applying the remainder formula to a sufficiently large prime number and check that the results exceeds our target I (10^{10} in this case). So, I guessed by using a list of primes at the 22044^{th} prime number 249,989 with a remainder of 2*22044*249989 = 11021515032. That’s greater than 10^{10} and will suffice.
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]]>The post Project Euler 128 Solution appeared first on Dreamshire.
]]>A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting at "12 o’clock" and numbering the tiles 2 to 7 in an anti-clockwise direction.
New rings are added in the same fashion, with the next rings being numbered 8 to 19, 20 to 37, 38 to 61, and so on. The diagram below shows the first three rings.
By finding the difference between tile n and each its six neighbours we shall define PD(n) to be the number of those differences which are prime.
For example, working clockwise around tile 8 the differences are 12, 29, 11, 6, 1, and 13. So PD(8) = 3.
In the same way, the differences around tile 17 are 1, 17, 16, 1, 11, and 10, hence PD(17) = 2.
It can be shown that the maximum value of PD(n) is 3.
If all of the tiles for which PD(n) = 3 are listed in ascending order to form a sequence, the 10th tile would be 271.
Find the 2000th tile in this sequence.
This is a modified version of the solution for probelm 58 – The square spiral – as this is a hex spiral constructed in the same way. As you may have guessed, just like ignoring the bottom-right diagonal in the square spiral, there are many directions we can ignore here as well as they always yield a non-prime difference (both even or both odd).
The solution below is a first swing at finding a solution. The real task is to learn of the generating function that will replace this brute force search.
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