他の方と解き方が同じで投稿するほどでもないかと思っていたのですが、何か参考になることもあるかと思い残しておきます。

問題 : http://nabetani.sakura.ne.jp/hena/ordf10updown/
実装リンク集 : https://qiita.com/Nabetani/items/dae474bc17827f9b1537

def number2yx(number):
    b = int((number - 1) ** 0.5)
    start = b ** 2 + 1
    a = min(number - start, start + b * 2 - number)
    return (a, b) if (b % 2 == 1) ^ (a == number - start) else (b, a)

def yx2number(y, x):
    if y < 0 or x < 0: return '-'
    a, b = sorted((y, x))
    start = b ** 2 + 1
    return start + (a if (b % 2 == 1) ^ (a == y) else b * 2 - a)

def solve(data):
    y, x = number2yx(int(data))
    return ','.join(str(yx2number(y + dy, x + dx))
                    for dy, dx in ((-1, 0), (1, 0), (0, -1), (0, 1)))

def test(data, expect):
    answer = solve(data)
    print("%s %s %s %s" % (answer == expect and 'OK' or 'NG', data, expect, answer))

test( "1", "-,2,-,4" );
test( "10", "9,25,-,11" );
test( "1", "-,2,-,4" );
test( "2", "1,9,-,3" );
test( "4", "-,3,1,5" );
test( "26", "25,49,-,27" );
test( "43", "42,56,44,58" );
test( "72", "71,73,57,93" );
test( "82", "81,121,-,83" );
test( "91", "92,112,90,110" );
test( "100", "-,99,65,101" );
test( "122", "121,169,-,123" );
test( "141", "142,140,104,148" );
test( "145", "-,146,144,196" );
test( "320", "321,319,261,329" );
test( "504", "503,505,465,557" );
test( "563", "564,562,498,590" );
test( "906", "905,907,895,1019" );
test( "1047", "1046,1048,1002,1134" );
test( "1111", "1068,1204,1110,1112" );
test( "1338", "1257,1401,1339,1337" );
test( "1613", "1612,1614,1588,1752" );
test( "1845", "1686,1854,1846,1844" );
test( "1921", "1922,1920,1780,1952" );
test( "2517", "2516,2518,2484,2688" );
test( "2670", "2671,2669,2535,2739" );
test( "2798", "2613,2821,2799,2797" );
test( "2841", "2778,2994,2840,2842" );
test( "2896", "2897,2895,2725,2937" );
test( "3050", "3001,3225,3049,3051" );
test( "3354", "3355,3353,3147,3375" );
test( "3563", "3564,3562,3402,3638" );
test( "4454", "4261,4525,4455,4453" );
test( "5397", "5262,5558,5396,5398" );
test( "5592", "5363,5659,5593,5591" );
test( "6122", "6121,6123,6047,6363" );
test( "6772", "6771,6773,6677,7009" );