Task: First perfect square in base n with n unique digits


C++: #include <string>
#include <iostream> 
#include <cstdlib>
#include <math.h>
#include <chrono>
#include <iomanip>

using namespace std;

const int maxBase = 16;  
int base, bmo, tc; 
const string chars = "0123456789ABCDEF"; 
unsigned long long full; 


string toStr(const unsigned long long ull) {
	unsigned long long u = ull; string res = ""; while (u > 0) {
		lldiv_t result1 = lldiv(u, base); res = chars[(int)result1.rem] + res;
		u = (unsigned long long)result1.quot;
	} return res;
}


unsigned long long to10(string s) {
	unsigned long long res = 0; for (char i : s) res = res * base + chars.find(i); return res;
}


bool allIn(const unsigned long long ull) {
	unsigned long long u, found; u = ull; found = 0; while (u > 0) {
		lldiv_t result1 = lldiv(u, base); found |= (unsigned long long)1 << result1.rem;
		u = result1.quot;
	} return found == full;
}


string fixup(int n) {
	string res = chars.substr(0, base); if (n > 0) res = res.insert(n, chars.substr(n, 1));
	return "10" + res.substr(2);
}


void doOne() {
	bmo = base - 1; tc = 0; unsigned long long sq, rt, dn, d;
	int id = 0, dr = (base & 1) == 1 ? base >> 1 : 0, inc = 1, sdr[maxBase] = { 0 };
	full = ((unsigned long long)1 << base) - 1;
	int rc = 0; for (int i = 0; i < bmo; i++) {
		sdr[i] = (i * i) % bmo; if (sdr[i] == dr) rc++; if (sdr[i] == 0) sdr[i] += bmo;
	}
	if (dr > 0) {
		id = base; for (int i = 1; i <= dr; i++)
			if (sdr[i] >= dr) if (id > sdr[i]) id = sdr[i]; id -= dr;
	}
	sq = to10(fixup(id)); rt = (unsigned long long)sqrt(sq) + 1; sq = rt * rt;
	dn = (rt << 1) + 1; d = 1; if (base > 3 && rc > 0) {
		while (sq % bmo != dr) { rt += 1; sq += dn; dn += 2; } 
		inc = bmo / rc; if (inc > 1) { dn += rt * (inc - 2) - 1; d = inc * inc; }
		dn += dn + d;
	} d <<= 1;
	do { if (allIn(sq)) break; sq += dn; dn += d; tc++; } while (true);
	rt += tc * inc;
	cout << setw(4) << base << setw(3) << inc << "  " << setw(2)
		<< (id > 0 ? chars.substr(id, 1) : " ") << setw(10) << toStr(rt) << "  "
		<< setw(20) << left << toStr(sq) << right << setw(12) << tc << endl;
}

int main() {
	cout << "base inc id      root  sqr                   test count" << endl;
	auto st = chrono::system_clock::now();
	for (base = 2; base <= maxBase; base++) doOne();
	chrono::duration<double> et = chrono::system_clock::now() - st;
	cout << "\nComputation time was " << et.count() * 1000 << " milliseconds" << endl;
	return 0;
}






Python: 

from itertools import count, dropwhile, repeat
from math import ceil, sqrt
from time import time



def allDigitSquare(base, above):
    
    bools = list(repeat(True, base))
    return next(
        dropwhile(
            missingDigitsAtBase(base, bools),
            count(
                max(
                    above,
                    ceil(sqrt(int(
                        '10' + '0123456789abcdef'[2:base],
                        base
                    )))
                )
            )
        )
    )



def missingDigitsAtBase(base, bools):
    
    def go(x):
        xs = bools.copy()
        while x:
            xs[x % base] = False
            x //= base
        return any(xs)
    return lambda n: go(n * n)



def digit(n):
    
    return '0123456789abcdef'[n]




def main():
    

    start = time()

    print(main.__doc__ + ':\n\nBase      Root    Square')
    q = 0
    for b in enumFromTo(2)(16):
        q = allDigitSquare(b, q)
        print(
            str(b).rjust(2, ' ') + ' -> ' +
            showIntAtBase(b)(digit)(q)('').rjust(8, ' ') +
            ' -> ' +
            showIntAtBase(b)(digit)(q * q)('')
        )

    print(
        '\nc. ' + str(ceil(time() - start)) + ' seconds.'
    )





def enumFromTo(m):
    
    return lambda n: list(range(m, 1 + n))




def showIntAtBase(base):
    
    def wrap(toChr, n, rs):
        def go(nd, r):
            n, d = nd
            r_ = toChr(d) + r
            return go(divmod(n, base), r_) if 0 != n else r_
        return 'unsupported base' if 1 >= base else (
            'negative number' if 0 > n else (
                go(divmod(n, base), rs))
        )
    return lambda toChr: lambda n: lambda rs: (
        wrap(toChr, n, rs)
    )



if __name__ == '__main__':
    main()