From b0d1b49fb45b86d204622b022d4675d7fa6fd6a9 Mon Sep 17 00:00:00 2001
From: Tibor Bizjak <tibor.bizjak@gmail.com>
Date: Sun, 26 Mar 2023 17:56:21 +0200
Subject: [PATCH] Added solutions for N=100-110

---
 100_arranged_probability.py        |   13 +
 101_optimum_polynomials.py         |   27 +
 102_triangle_containment.py        |   23 +
 103_special_subset_sums_optimum.py |   46 ++
 104_pandigital_fibonacci_ends.py   |   25 +
 105_special_subset_sums_testing.py |   21 +
 106_special_subset_sums_meta.py    |   17 +
 107_minimal_network.py             |   30 +
 108_diophantine_reciprocals_I.py   |   54 ++
 109_darts.py                       |   24 +
 data/102_triangles.txt             | 1000 ++++++++++++++++++++++++++++
 data/105_sets.txt                  |  100 +++
 data/107_network.txt               |   40 ++
 data/107_test.txt                  |    7 +
 14 files changed, 1427 insertions(+)
 create mode 100644 100_arranged_probability.py
 create mode 100644 101_optimum_polynomials.py
 create mode 100644 102_triangle_containment.py
 create mode 100644 103_special_subset_sums_optimum.py
 create mode 100644 104_pandigital_fibonacci_ends.py
 create mode 100644 105_special_subset_sums_testing.py
 create mode 100644 106_special_subset_sums_meta.py
 create mode 100644 107_minimal_network.py
 create mode 100644 108_diophantine_reciprocals_I.py
 create mode 100644 109_darts.py
 create mode 100644 data/102_triangles.txt
 create mode 100644 data/105_sets.txt
 create mode 100644 data/107_network.txt
 create mode 100644 data/107_test.txt

diff --git a/100_arranged_probability.py b/100_arranged_probability.py
new file mode 100644
index 0000000..39eda9b
--- /dev/null
+++ b/100_arranged_probability.py
@@ -0,0 +1,13 @@
+#!/usr/bin/env python3
+
+lim = 10**12
+x, y = 3, 1
+blue = 3
+total  = 4
+
+while total <= lim:
+    x, y = 3*x + 8*y, 3*y + x
+    blue = (2*y + x + 1) // 2
+    total = (4*y + x + 1) // 2
+
+print(blue)
diff --git a/101_optimum_polynomials.py b/101_optimum_polynomials.py
new file mode 100644
index 0000000..3c270d6
--- /dev/null
+++ b/101_optimum_polynomials.py
@@ -0,0 +1,27 @@
+#!/usr/bin/env python3
+
+CFS = [1, -1]*5 + [1]
+
+def main(cfs):
+    seq = getseq(cfs)
+    py  = pyramid(seq)
+    return sum(op(n, py) for n in range(1, len(cfs)))
+
+def getseq(cfs):
+    return [sum(c*n**pw for pw, c in enumerate(cfs))
+            for n in range(1, len(cfs))]
+
+def pyramid(seq):
+    r = [seq]
+    for i in range(len(seq)-1):
+        lvl = []
+        for j in range(len(r[i])-1):
+            lvl.append(r[i][j+1] - r[i][j])
+        r.append(lvl)
+    return r
+
+def op(n, py):
+    return sum(py[i][n-i-1] for i in range(n))
+
+#print(main([0,0,0,1]))
+print(main(CFS))
diff --git a/102_triangle_containment.py b/102_triangle_containment.py
new file mode 100644
index 0000000..fe73595
--- /dev/null
+++ b/102_triangle_containment.py
@@ -0,0 +1,23 @@
+#!/usr/bin/env python3
+
+# Calculate area of triangle and compare to areas
+# of triangles with one vertex = origin
+# better solution -> check sign of cross products
+
+def main():
+    tr = parse("data/102_triangles.txt")
+    return len([0 for p in tr if areaTR(p) == areaOG(0,2,p) + areaOG(0,4,p) + areaOG(2,4,p)])
+
+def areaTR(p):  #triangle area
+    return abs(p[0]*p[3] - p[0]*p[5] + 
+               p[2]*p[5] - p[2]*p[1] +
+               p[4]*p[1] - p[4]*p[3])
+
+def areaOG(a, b, p): #triangle with with a=0,0
+    return abs(p[a]*p[b+1] - p[b]*p[a+1])
+
+def parse(path):
+    with open(path) as f:
+        return [list(map(int, row.split(","))) for row in f.read()[:-1].split("\n")]
+
+print(main())
diff --git a/103_special_subset_sums_optimum.py b/103_special_subset_sums_optimum.py
new file mode 100644
index 0000000..256eba0
--- /dev/null
+++ b/103_special_subset_sums_optimum.py
@@ -0,0 +1,46 @@
+#!/usr/bin/env python3
+
+def test(n, setsums):
+    newsums = [s+n for s in setsums] + setsums + [n]
+    if len(newsums) != len(set(newsums)):
+        return False
+    return newsums
+
+def search(left, right, length, best, setsums, lor, w): #left to right
+    if len(left + right) >= length:
+        if sum(left + right) < best:
+            #print left+right, sum(left+right)
+            for i, n in enumerate(left+right):
+                w[i] = n
+            return sum(left + right)
+        return best
+    hole = length - len(left + right) - 1
+    minsum = sum(left + right) + (hole*(hole+1))//2 + left[-1]*hole
+    if lor:
+        n = max(left[-1]+1, sum(right)-sum(left) + 1)
+        bound = right[0] - hole
+    else:
+        n = left[-1] + hole + 1
+        bound = best - minsum
+        if len(right) > 0 and right[0] < bound:
+            bound = right[0]
+    while n < bound and n < best - minsum - (n-left[-1])*hole*lor:
+        newsums = test(n, setsums)
+        if newsums:
+            best = search(left + [n]*lor,
+                          [n]*(not lor) + right,
+                          length, best, newsums, not lor, w)
+        n += 1
+    return best
+
+def base(length, best):
+    x = length//2 + length % 2
+    n = (x + 1 - length%2) * (x-1) + 1
+    n = 20
+    w = [0]*length
+    while n < best - (length-1)*length - n*(length-1):
+        best = search([n], [], length, best, [n], False, w)
+        n += 1
+    return "".join(map(str, w)), best
+
+print(base(7, 500)[0])
diff --git a/104_pandigital_fibonacci_ends.py b/104_pandigital_fibonacci_ends.py
new file mode 100644
index 0000000..2e771bb
--- /dev/null
+++ b/104_pandigital_fibonacci_ends.py
@@ -0,0 +1,25 @@
+#!/usr/bin/env python3
+
+from math import sqrt, log10
+
+def main():
+    k = 1
+    m    = 0
+    last = 1
+    gr    = (1 + sqrt(5)) / 2
+    first = gr*(1/sqrt(5))
+    while cond(str(int(first))[:9], str(last)):
+        last, m = (last+m)%10**9, last
+        first = gr*first
+        if first >= 10**9:
+            first /= 10
+        k += 1
+    return k, int(log10(gr)*k - log10(1/sqrt(5)) + 1/2.0)
+
+def cond(first, last):
+    a = len(set(last))  != 9
+    b = len(set(first)) != 9
+    c = '0' in first+last
+    return a or b or c
+
+print(main()[0])
diff --git a/105_special_subset_sums_testing.py b/105_special_subset_sums_testing.py
new file mode 100644
index 0000000..b405a59
--- /dev/null
+++ b/105_special_subset_sums_testing.py
@@ -0,0 +1,21 @@
+#!/usr/bin/env python3
+
+def test(s):
+    if len(s) != len(set(s)):
+        return False
+    s.sort()
+    if sum(s[:len(s)//2 + len(s)%2]) <= sum(s[len(s)//2+1:]):
+        return False
+    setsums = []
+    for n in s:
+        setsums += [m+n for m in setsums] + [n]
+        if len(setsums) != len(set(setsums)):
+            return False
+    return True
+
+with open("data/105_sets.txt") as f:
+    r = f.read()
+
+sets = [list(map(int, line.split(','))) for line in r.split('\n')[:-1]]
+
+print(sum(sum(s) for s in sets if test(s)))
diff --git a/106_special_subset_sums_meta.py b/106_special_subset_sums_meta.py
new file mode 100644
index 0000000..c5174cb
--- /dev/null
+++ b/106_special_subset_sums_meta.py
@@ -0,0 +1,17 @@
+#!/usr/bin/env python3
+
+from math import factorial
+
+def bin(n, k):
+    if n < 0:
+        return 0
+    if n < k:
+        return 0
+    return factorial(n) // (factorial(n-k) * factorial(k))
+
+n = 12
+ms = 0
+for k in range(0, n//2 +1):
+    ms += bin(n, 2*k) * bin(2*k-1, k+1)
+
+print(ms)
diff --git a/107_minimal_network.py b/107_minimal_network.py
new file mode 100644
index 0000000..a27de4d
--- /dev/null
+++ b/107_minimal_network.py
@@ -0,0 +1,30 @@
+#!/usr/bin/env python3
+
+def parse(fname):
+    with open(fname) as f:
+        r = f.read()
+    r = [line.split(',') for line in r.split('\n')[:-1]]
+    weights = {a : [(b, int(w)) for b, w in enumerate(line)
+                                   if w!='-'
+                   ]
+                   for a, line in enumerate(r)
+              }
+    return weights
+
+def search(network, visited={0}):
+    if len(visited) == len(network):
+        return 0
+    edges = [(n2, w) for node in visited
+                   for n2, w in network[node]
+                   if n2 not in visited
+            ]
+    next, w = min(edges, key = lambda x: x[1])
+    return w + search(network, visited | {next})
+
+network = parse("data/107_network.txt")
+total_weight  = sum(w for node in network
+                     for _, w in network[node])
+total_weight //= 2
+min_weight    = search(network)
+
+print(total_weight - min_weight)
diff --git a/108_diophantine_reciprocals_I.py b/108_diophantine_reciprocals_I.py
new file mode 100644
index 0000000..1eb8b6d
--- /dev/null
+++ b/108_diophantine_reciprocals_I.py
@@ -0,0 +1,54 @@
+#!/usr/bin/env python3
+
+from lib import primegen
+from math import log
+from operator import mul
+from functools import reduce
+
+def calc(pows):
+    return (reduce(mul, (2*p + 1 for p in pows)) + 1) // 2
+
+def increment(i, pows):
+    n = 0
+    if i < len(pows):
+        n = pows[i]
+    return [n + 1] * (i+1) + pows[i+1:]
+
+def val(pows):
+    return sum(p*v for p, v in zip(pows, vals))
+
+def fpwof2(pows, n):
+    if len(pows) == 1:
+        return n-1
+    r = (2*n - 1) // reduce(mul, (2*p + 1 for p in pows[1:]))
+    r = (r - 1) // 2
+    while calc([r] + pows[1:]) < n:
+        r += 1
+    return r
+
+N = 10 ** 3
+
+primes = list(primegen(100))
+vals   = [log(p) / log(2) for p in primes]
+
+i = 0
+pows = [0]
+best = fpwof2(pows, N)
+best_pows = None
+
+while not (val(pows) >= best and sum(pows) == len(pows)):
+    if not i:
+        pows[0] = fpwof2(pows, N)
+        if len(pows) > 1 and pows[0] < pows[1]:
+            pows[0]  = pows[1]
+        if val(pows) < best:
+            best = val(pows)
+            best_pows = pows
+        i = 1
+    pows = increment(i, pows)
+    if val(pows) >= best:
+        i += 1
+    else:
+        i = 0
+
+print(reduce(mul, (n**pw for n, pw in zip(primes, best_pows))))
diff --git a/109_darts.py b/109_darts.py
new file mode 100644
index 0000000..1bdcb1c
--- /dev/null
+++ b/109_darts.py
@@ -0,0 +1,24 @@
+#!/usr/bin/env python3
+
+bull = [25]
+miss = [0]
+fields = list(range(1, 21))
+single = fields + bull
+double = [2*i for i in single] 
+treble  =[3*i for i in fields]
+
+regs = miss + single + double + treble
+regs.sort()
+
+count = 0
+score = 100
+
+for fst in double:
+    for i, sec in enumerate(regs):
+        if fst + 2*sec >= score:
+            break
+        for thd in regs[i:]:
+            if fst + sec + thd >= score:
+                break
+            count += 1
+print(count)
diff --git a/data/102_triangles.txt b/data/102_triangles.txt
new file mode 100644
index 0000000..3f01a1a
--- /dev/null
+++ b/data/102_triangles.txt
@@ -0,0 +1,1000 @@
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diff --git a/data/105_sets.txt b/data/105_sets.txt
new file mode 100644
index 0000000..2a9035a
--- /dev/null
+++ b/data/105_sets.txt
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diff --git a/data/107_network.txt b/data/107_network.txt
new file mode 100644
index 0000000..fcc3c41
--- /dev/null
+++ b/data/107_network.txt
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diff --git a/data/107_test.txt b/data/107_test.txt
new file mode 100644
index 0000000..f5f2acc
--- /dev/null
+++ b/data/107_test.txt
@@ -0,0 +1,7 @@
+-,16,12,21,-,-,-
+16,-,-,17,20,-,-
+12,-,-,28,-,31,-
+21,17,28,-,18,19,23
+-,20,-,18,-,-,11
+-,-,31,19,-,-,27
+-,-,-,23,11,27,-