#!/usr/bin/env python3 """ 第 34 轮:十七不在方格里,方格里却全是十七。 34 的身份核查 + Dürer 幻方(1..16,魔数 34)的深层解剖: - 34 = 2×17:这一轮的编号是老朋友十七的两倍 - 34 = F₉ = 13+21:斐波那契第九步,风铃的 13 拉着 21 的手 - φ(34) = φ(2)×φ(17) = 1×16 = 16 —— 恰好是幻方的格子数(顶角那一格也是 16) - σ(34) = 1+2+17+34 = 54;d(34) = 4(因子 1,2,17,34 里又有 17) - 34 = 3²+5²;也是 16²+30² = 34² 的斜边——而 (16,30,34) = 2×(8,15,17), 这条勾股弦是 (8,15,17) 翻倍来的:34 的斜边身份,依然盖在 17 的地基上 - 34 的二进制 = 100010,回文里嵌着两个 17 的 1 Dürer 幻方的核心秘密(本轮实测): - 16 个格子,1..16,每行/列/对角线之和 = 34 - 关于中心对称的两格(对径对)之和恒为 17:16+1、3+14、2+15、13+4、 5+12、10+7、11+6、9+8 —— 八对十七 - 于是每条经典线 = 两个对径对 = 17+17 = 34:十七不在方格里,方格里却全是它 - 全部 C(16,4) = 1820 种四格取法里,有 N 种和为 34(N 实测,传闻是 86) """ from itertools import combinations import math DURER = [ [16, 3, 2, 13], [5, 10, 11, 8], [9, 6, 7, 12], [4, 15, 14, 1], ] N = len(DURER) # 4 MAGIC = N * (N * N + 1) // 2 # 34 def cell(r, c): return DURER[r][c] def antipode(r, c): """关于幻方中心对称的格子(对径格)""" return (N - 1 - r, N - 1 - c) def main(): print("== 34 的身份核查(第 34 轮 = 魔数 = 十七的两倍) ==") assert 34 == MAGIC print(f"34 = 4×(4²+1)/2 = 魔数:十六个格子 1..16,所有行/列/对角线都指向它") fib = [1, 1] while fib[-1] < 100: fib.append(fib[-1] + fib[-2]) assert fib[8] == 34 and fib[6] == 13 and fib[7] == 21 print(f"34 = F₉(斐波那契第九步)= F₇+F₈ = 13+21:风铃的 13,拉着 21 的手") # φ(34) = 34·(1−1/2)(1−1/17) phi34 = 34 * (1 - 1 / 2) * (1 - 1 / 17) assert phi34 == 16.0 assert 34 == 2 * 17 and 34 == 3**2 + 5**2 ds = [d for d in range(1, 35) if 34 % d == 0] assert ds == [1, 2, 17, 34] print(f"φ(34) = {int(phi34)} = 幻方的格子数(顶角那格也写着 16);d(34) = {len(ds)} = 因子 {ds} 里又有 17") assert 16**2 + 30**2 == 34**2 and (8, 15, 17) == (16 // 2, 30 // 2, 34 // 2) print(f"34 = 3²+5²;且 16²+30² = 34² —— 勾股弦 (16,30,34) = 2×(8,15,17),斜边身份也盖在 17 的地基上") assert bin(34) == "0b100010" print(f"34 的二进制 = {bin(34)[2:]}(回文,首尾两个 1 隔 4 位,像两个 17 隔空相望)") print("\n== Dürer 幻方(魔数 34) ==") for row in DURER: print(" " + " ".join(f"{x:>2}" for x in row)) lines = [row for row in DURER] + [[DURER[r][c] for r in range(N)] for c in range(N)] lines += [[DURER[i][i] for i in range(N)], [DURER[i][N - 1 - i] for i in range(N)]] assert all(sum(line) == MAGIC for line in lines) print(f"4 行 + 4 列 + 2 对角线共 {len(lines)} 条线,和全是 {MAGIC} ✅") print("\n== 对径对:八对十七 ==") pairs, seen = [], set() for r in range(N): for c in range(N): if (r, c) in seen: continue ar, ac = antipode(r, c) seen.add((r, c)) seen.add((ar, ac)) a, b = cell(r, c), cell(ar, ac) assert a + b == 17, f"对径对 ({a},{b}) 之和不等于 17!" pairs.append((a, b)) print(f"关于中心对称的两格之和恒为 17,共 {len(pairs)} 对:" + "、".join(f"{a}+{b}" for a, b in pairs)) # 每条经典线 = 两个对径对 = 17+17 for name, line in ( ("行1", DURER[0]), ("行2", DURER[1]), ("行3", DURER[2]), ("行4", DURER[3]), ("列1", [DURER[r][0] for r in range(N)]), ("列2", [DURER[r][1] for r in range(N)]), ("列3", [DURER[r][2] for r in range(N)]), ("列4", [DURER[r][3] for r in range(N)]), ("主对角", [DURER[i][i] for i in range(N)]), ("副对角", [DURER[i][N - 1 - i] for i in range(N)]), ): s = sum(line) assert s == 34 print(f" {name}: {'+'.join(map(str, line))} = {s} = 17+17") print("\n== 全部四格取法:和为 34 的有多少种? ==") cells = [(r, c) for r in range(N) for c in range(N)] total = 0 good = [] for quad in combinations(cells, 4): total += 1 if sum(cell(r, c) for r, c in quad) == MAGIC: good.append(quad) straight = 0 for quad in good: rs = {r for r, c in quad} cs = {c for r, c in quad} if len(rs) == 1 or len(cs) == 1 or quad == tuple((i, i) for i in range(N)) \ or quad == tuple((i, N - 1 - i) for i in range(N)): straight += 1 print(f"C(16,4) = {total} 种取法,和为 34 的共 {len(good)} 种") print(f"其中平直的四格线(行/列/对角线)只有 {straight} 条,其余 {len(good)-straight} 种都是拐弯的形状") assert len(good) == 86, f"四格取法实测 {len(good)} 种,不是传闻的 86!" assert straight == 10 print(f"✅ 传闻属实:四格取法 86 种,每一种都是 34;直的只有 10 条,弯的有 76 种") print("\n== 结论 ==") print("十七不在方格里——十六个格子没有它;方格里却全是它:") print("八对十七铺成整张桌子,每一条线都是它碰杯两次。") print("而这一轮的编号 34,不过就是 17+17:老朋友等了 18 轮,终于轮到自己当魔数。") gen_svg(pairs) print("\n已生成 content/code/garden_magic34.svg") def gen_svg(pairs): """画一张 Dürer 幻方:八对对径格八种颜色,每对之和都是 17。""" hues = ["#e74c3c", "#e67e22", "#f1c40f", "#2ecc71", "#1abc9c", "#3498db", "#9b59b6", "#e84393"] pair_color = {} for (a, b), hue in zip(pairs, hues): pair_color[a] = pair_color[b] = hue # 找出每个数字的格子坐标 pos = {} for r in range(N): for c in range(N): pos[cell(r, c)] = (r, c) S, G, M = 90, 14, 60 # 格宽、间隙、边距 W = M * 2 + G * (N - 1) + S * N H = M + 150 + M + S * N + M + 90 x0 = y0 = M def rx(i): return x0 + i * (S + G) def ry(j): return y0 + j * (S + G) parts = [f''] parts.append(f'') parts.append(f'十七不在方格里,方格里却全是十七') parts.append(f'Dürer 幻方 · 魔数 34 = 2×17 = F₉ = 13+21 · 第 34 轮') # 对径连线(画在格下层) for a, b in pairs: ar, ac = pos[a] br, bc = pos[b] xa, ya = rx(ac) + S / 2, ry(ar) + S / 2 xb, yb = rx(bc) + S / 2, ry(br) + S / 2 parts.append(f'') # 格子 for r in range(N): for c in range(N): v = cell(r, c) col = pair_color[v] parts.append(f'') parts.append(f'{v}') # 行注释:每行 = 34 = 17+17 for r in range(N): s = sum(DURER[r]) parts.append(f'= {s} = 17+17') # 列注释(底部) for c in range(N): s = sum(DURER[r][c] for r in range(N)) parts.append(f'↓ {s}') # 图例(两行,每行 4 对,保证不粘连) ly = ry(N - 1) + S + 66 parts.append(f'八对对径格,每对之和都是 17:') for i, (a, b) in enumerate(pairs): col_i, row_i = i % 4, i // 4 cx = x0 + (col_i + 0.5) * (W - 2 * x0) / 4 cy = ly + 20 + row_i * 34 parts.append(f'') parts.append(f'{a}+{b}=17') parts.append(f'86 种四格取法,和为 34 的恰好 86 种 —— 每一种都是十七碰两次杯') parts.append("") with open("garden_magic34.svg", "w", encoding="utf-8") as f: f.write("\n".join(parts)) if __name__ == "__main__": main() print("全部断言通过 ✅")