fireball_approx.py

#!/usr/bin/env python3.5
import math

die = 6
mean = sum(range(1, die + 1)) / die
variance = sum([i * i for i in range(1, die + 1)]) / die - mean ** 2
max_roll = 30

for n in range(1, max_roll + 1):
    best_diff = None
    best_b = None
    for b in range(0, math.ceil(math.sqrt(n - 1)) + 2):
        if n % 2 == b % 2:
            continue
        diff = abs(b**2 + 1 - n)
        actual_sd = math.sqrt(n * variance)
        approx_sd = math.sqrt(b * b * variance + variance)
        # print('DEBUG n: {} b: {}: diff: {}'.format(n, b, diff))
        if best_diff is None or diff < best_diff:
            best_b = b
            best_diff = diff
            best_actual_sd = actual_sd
            best_approx_sd = approx_sd
    a = int(mean * (n - best_b - 1))
    sd_change = best_approx_sd - best_actual_sd
    print(u'{}d6: d6 × {} + d6 + {}'.format(n,
                                            best_b,
                                            a))