p128.py 7.11 KB
 Nayuki Minase committed Jul 28, 2015 1 2 ``````# # Solution to Project Euler problem 128 `````` Nayuki committed Dec 27, 2016 3 ``````# Copyright (c) Project Nayuki. All rights reserved. `````` Nayuki Minase committed Jul 28, 2015 4 ``````# `````` Nayuki Minase committed Feb 05, 2016 5 ``````# https://www.nayuki.io/page/project-euler-solutions `````` Nayuki Minase committed Jul 28, 2015 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 ``````# https://github.com/nayuki/Project-Euler-solutions # import eulerlib, itertools # Let's do mathematical analysis to drastically reduce the amount of # logic we need to implement and calculation the computer needs to do. # We begin with a couple of definitions. # # Ring number: Each cell belongs in a hexagonal ring, # numbered starting from 0 at the center like this: # 3 # 3 3 # 3 2 3 # 3 2 2 3 # 2 1 2 # 3 1 1 3 # 2 0 2 # 3 1 1 3 # 2 1 2 # 3 2 2 3 # 3 2 3 # 3 3 # 3 # # Corner/edge cell: Within a ring, each cell is # either a corner cell or an edge cell, as shown: # C # E E # E C E # C E E C # C C C # E C C E # E C E # E C C E # C C C # C E E C # E C E # E E # C # # Basic observations: # - Except for the degenerate ring 0, each ring k has 6k cells. # The kth ring has exactly 6 corner cells and 6(k - 1) edge cells. # - In the code we will skip the PD (prime difference) calculation for # rings 0 and 1 because the existence of ring 0 breaks many patterns. # - Doing the PD calculation for rings 0 and 1 by hand (n = 1 to 7 # inclusive), we find that PD(n) = 3 for and only for n = 1, 2. # # Now let's analyze the characteristics of all cells in rings 2 or above. # It's hard to justify these assertions rigorously, but they are true from # looking at the spiral diagram. # # - Corner cells along the upward vertical direction and the edge cells # immediately to the right of this vertical column are the most interesting, # so we will save these cases for last. # # - Claim: Except for cells immediately right of the upward corner column, # no edge cell satisfies PD(n) = 3. Proof: Take an arbitrary edge cell n # not immediately to the right of the upward corner column... # - The two neighbors in the same ring have a difference of 1 compared to n, # which is not a prime number. # - The two neighbors in the previous (inward) ring are consecutive numbers, # so exactly one of them has an even absolute difference with n. Because # n is in ring 2 or above, the difference with any neighboring number in the # previous ring is at least 6. Thus an even number greater than 2 is not prime. # - Similarly, the two neighbors in the next (outward) ring are consecutive numbers. # One of them has an even difference with n, and this number is also at least 6, # so one neighbor is definitely not prime. # - Therefore with at least 4 neighbors that do not have a prime difference, PD(n) <= 2. # Example of an edge cell n = 11 in ring 2, which is straight left of the origin: # 10 # 24 03 # 11 # 25 04 # 12 # # - Claim: No corner cell in the other 5 directions satisfies PD(n) = 3. # Proof: Take an arbitrary corner cell n in the non-upward direction... # - Two of its neighbors (in the same ring) have a difference of 1, # which is not prime. # - One neighbor is in the previous ring (inward) while three neighbors # are in the next ring (outward). # - Let the inner ring neighbor be k and the outer ring's middle neighbor # be m. The three outer ring neighbors are {m - 1, m, m + 1}. # - Then n - k + 6 = m - n. Also, {m - 1, m + 1} have the same parity, # and {k, m} have the same other parity. # - Either both {|k - n|, |m - n|} are even or both {|m - 1 - n|, |m + 1 - n|} are even. # In any case, all these differences are at least 6, so the even numbers are not prime. # - Therefore with at least 4 neighbors that do not have a prime difference, PD(n) <= 2. # Example of a corner cell n = 14 in ring 2, which is straight below the origin: # 05 # 13 15 # 14 # 28 30 # 29 # # - Now let's consider an arbitrary upward corner cell n in ring k, with k >= 2. # We shall give variables to all its neighbors like this: # d # e f # n # b c # a # - a is in the previous ring, {b, c} are in the same ring as n, # and {d, e, f} are in the next ring. # - Equations derived from the structure of the hexagonal spiral: # n = 3k(k - 1) + 2. # a = n - 6(k - 1). # b = n + 1. # c = n + 6k - 1 = d - 1. # d = n + 6k. # e = n + 6k + 1 = d + 1. # f = n + 6k + 6(k + 1) - 1 = n + 12k + 5. # - Hence we get these absolute differences with n: # |a - n| = 6(k - 1). (Not prime because it's a multiple of 6) # |b - n| = 1. (Not prime) # |c - n| = 6k - 1. (Possibly prime) # |d - n| = 6k. (Not prime because it's a multiple of 6) # |e - n| = 6k + 1. (Possibly prime) # |f - n| = 12k + 5. (Possibly prime) # - Therefore for each k >= 2, we need to count how many numbers # in the set {6k - 1, 6k + 1, 12k + 5} are prime. # Example of a corner cell n = 8 in ring 2, which is straight above the origin: # 20 # 21 37 # 08 # 09 19 # 02 # # - Finally let's consider an arbitrary edge cell immediately to the right of the # upward vertical column. Suppose the cell's value is n and it is in ring k, # with k >= 2. Give variables to all its neighbors like this: # f # c e # n # a d # b # - {a, b} are in the previous ring, {c, d} are in the current ring, and {e, f} are in # the next ring. The ascending ordering of all these numbers is (a, b, c, d, n, e, f). # - Equations derived from the structure of the hexagonal spiral: # n = 3k(k + 1) + 1. # a = n - 6k - 6(k - 1) + 1 = n - 12k + 7. # b = n - 6k. # c = n - 6k + 1. # d = n - 1. # e = n + 6(k + 1) - 1 = n + 6k + 5. # f = n + 6(k + 1). # - Hence we get these absolute differences with n: # |a - n| = 12k - 7. (Possibly prime) # |b - n| = 6k. (Not prime because it's a multiple of 6) # |c - n| = 6k - 1. (Possibly prime) # |d - n| = 1. (Not prime) # |e - n| = 6k + 5. (Possibly prime) # |f - n| = 6(k + 1). (Not prime because it's a multiple of 6) # - Therefore for each k >= 2, we need to count how many numbers # in the set {6k - 1, 6k + 5, 12k - 7} are prime. # Example of an edge cell n = 19 in ring 2: # 37 # 08 36 # 19 # 02 18 # 07 def compute(): TARGET = 2000 # Must be at least 3 count = 2 # Because n = 1 and 2 satisfy PD(n) = 3 for ring in itertools.count(2): `````` Nayuki Minase committed Aug 13, 2015 174 `````` if all(map(eulerlib.is_prime, (ring * 6 - 1, ring * 6 + 1, ring * 12 + 5))): `````` Nayuki Minase committed Jul 28, 2015 175 176 177 `````` count += 1 if count == TARGET: return str(ring * (ring - 1) * 3 + 2) `````` Nayuki Minase committed Aug 13, 2015 178 `````` if all(map(eulerlib.is_prime, (ring * 6 - 1, ring * 6 + 5, ring * 12 - 7))): `````` Nayuki Minase committed Jul 28, 2015 179 180 181 182 183 184 185 `````` count += 1 if count == TARGET: return str(ring * (ring + 1) * 3 + 1) if __name__ == "__main__": print(compute())``````