Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
| Triangle | \(P_{3,n}=n(n+1)/2\) | 1, 3, 6, 10, 15, … |
| Square | \(P_{4,n}=n2\) | 1, 4, 9, 16, 25, … |
| Pentagonal | \(P_{5,n}=n(3n−1)/2\) | 1, 5, 12, 22, 35, … |
| Hexagonal | \(P_{6,n}=n(2n−1)\) | 1, 6, 15, 28, 45, … |
| Heptagonal | \(P_{7,n}=n(5n−3)/2\) | 1, 7, 18, 34, 55, … |
| Octagonal | \(P_{8,n}=n(3n−2)\) | 1, 8, 21, 40, 65, … |
The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.
- The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
- Each polygonal type: triangle (\(P_{3,127}=8128\)), square (\(P_{4,91}=8281\)), and pentagonal (\(P_{5,44}=2882\)), is represented by a different number in the set.
- This is the only set of 4-digit numbers with this property.
Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set. 一开始因为公式用错了，算出来很多值，都有点怀疑人生了。这个对计算量要求很低的感觉。
import itertools
TRANGLE_INDEX = {}
SQUAEW_INDEX = {}
PENTAGONAL_INDEX = {}
HEXAGONAL_INDEX = {}
HEPTAGONAL_INDEX = {}
OCTAGONAL_INDEX = {}
def triangle(number):
return number*(number+1) / 2
def square(number):
return number*number
def pentagonal(number):
return number*(3*number - 1)/2
def hexagonal(number):
return number*(2*number - 1)
def heptagonal(number):
return number*(5*number - 3) / 2
def octagonal(number):
return number*(3*number - 2)
for i in range(1, 1000):
result = triangle(i)
if result < 1000:
continue
if result > 9999:
break
else:
if not str(result)[2] == '0':
TRANGLE_INDEX[i] = result
for i in range(1, 1000):
result = square(i)
if result < 1000:
continue
if result > 9999:
break
else:
if not str(result)[2] == '0':
SQUAEW_INDEX[i] = result
for i in range(1, 1000):
result = pentagonal(i)
if result < 1000:
continue
if result > 9999:
break
else:
if not str(result)[2] == '0':
PENTAGONAL_INDEX[i] = result
for i in range(1, 1000):
result = hexagonal(i)
if result < 1000:
continue
if result > 9999:
break
else:
if not str(result)[2] == '0':
HEXAGONAL_INDEX[i] = result
for i in range(1, 1000):
result = heptagonal(i)
if result < 1000:
continue
if result > 9999:
break
else:
if not str(result)[2] == '0':
HEPTAGONAL_INDEX[i] = result
for i in range(1, 1000):
result = octagonal(i)
if result < 1000:
continue
if result > 9999:
break
else:
if not str(result)[2] == '0':
OCTAGONAL_INDEX[i] = result
# print TRANGLE_INDEX
# print
# print SQUAEW_INDEX
# print
# print PENTAGONAL_INDEX
# print
# print HEXAGONAL_INDEX
# print
# print HEPTAGONAL_INDEX
# print
# print OCTAGONAL_INDEX
o = itertools.permutations([TRANGLE_INDEX, SQUAEW_INDEX, PENTAGONAL_INDEX,
HEXAGONAL_INDEX, HEPTAGONAL_INDEX, OCTAGONAL_INDEX], 6)
for a, b, c, d, e, f in o:
order = [a, b, c, d, e, f]
for k_1 in order[0]:
v_1 = order[0][k_1]
v_1_t = str(v_1)[2] + str(v_1)[3]
v_1_h = str(v_1)[0] + str(v_1)[1]
for k_2 in order[1]:
v_2 = order[1][k_2]
v_2_t = str(v_2)[2] + str(v_2)[3]
v_2_h = str(v_2)[0] + str(v_2)[1]
if v_2_h == v_1_t and not k_1 == k_2:
for k_3 in order[2]:
v_3 = order[2][k_3]
v_3_t = str(v_3)[2] + str(v_3)[3]
v_3_h = str(v_3)[0] + str(v_3)[1]
if v_3_h == v_2_t and not k_1 == k_3 and not k_2 == k_3:
for k_4 in order[3]:
v_4 = order[3][k_4]
v_4_t = str(v_4)[2] + str(v_4)[3]
v_4_h = str(v_4)[0] + str(v_4)[1]
if v_4_h == v_3_t and not k_1 == k_4 and not k_2 == k_4 and not k_3 == k_4:
for k_5 in order[4]:
v_5 = order[4][k_5]
v_5_t = str(v_5)[2] + str(v_5)[3]
v_5_h = str(v_5)[0] + str(v_5)[1]
if v_5_h == v_4_t and not k_1 == k_5 and not k_2 == k_5 and not k_3 == k_5 and not k_4 == k_5:
for k_6 in order[5]:
v_6 = order[5][k_6]
v_6_t = str(v_6)[2] + str(v_6)[3]
v_6_h = str(v_6)[0] + str(v_6)[1]
if v_6_h == v_5_t and not k_1 == k_6 and not k_2 == k_6 and not k_3 == k_6 and not k_4 == k_6 and not k_5 == k_6:
if v_1_h == v_6_t:
print "%s(%s) %s(%s) %s(%s) %s(%s) %s(%s) %s(%s)" % (v_1, k_1, v_2, k_2, v_3, k_3, v_4, k_4, v_5, k_5, v_6, k_6)
print v_1 + v_2 + v_3 + v_4 + v_5 + v_6