Euler – Coder.si

Cyclical figurate numbers

Problem 61

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:

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.

# 45 - 141
def triangle(n):
    return n * (n + 1) / 2


# 32 - 101
def square(n):
    return n**2


# 26 - 82
def pentagonal(n):
    return n * (3 * n - 1) / 2


# 23 - 71
def hexagonal(n):
    return n * (2 * n - 1)


# 21 - 64
def heptagonal(n):
    return n * (5 * n - 3) / 2


# 19 - 59
def octagonal(n):
    return n * (3 * n - 2)


ranges = [(45, 141), (32, 101), (26, 82), (23, 71), (21, 64), (19, 59)]
funs = [triangle, square, pentagonal, hexagonal, heptagonal, octagonal]


def build_map():
    data = {}
    for i in range(6):
        fun = funs[i]
        for j in range(ranges[i][0], ranges[i][1]):
            v = fun(j)
            p = str(v / 100)
            # the value map under this prefix
            vs = data.setdefault(p, {})

            # the value map for the specified prefix and under current index
            vis = vs.setdefault(i, [])
            vis.append(v)
    return data


data = build_map()


def find(idx_list, value, result):
    r = value % 100
    if r > 10:
        prefix = str(r)
        if len(idx_list) == 6:
            if str(result[0] / 100) == prefix:
                # print idx_list, result
                print sum(result)
        else:
            # get value map under the prefix
            vs = data.setdefault(prefix, {})

            # the value map for the specified prefix and under current index
            for idx, vis in vs.items():
                if not idx in idx_list:
                    for vi in vis:
                        find(idx_list + [idx], vi, result + [vi])


def run():
    for i in range(ranges[5][0], ranges[5][1]):
        v = funs[5](i)
        find([5], v, [v])