Find the smallest 10-digit number that is a perfect square and contains all the digits 0 to 9.

With a program, this problem is solved easily.

Python3
from math import sqrt
# only consider the values from 1,000,000,000 to 10,000,000,000
Start = int(sqrt(1e9)) + 1
End = int(sqrt(1e10))
for n in range(Start, End):
    if len(set(str(n * n))) == 10:
        print(n * n)
        break

$\text{list}=\text{Select}[\text{Table}[\text{FromDigits}[p],\{p,\text{Permutations}[\text{Range}[0,9]]\}],\text{\$\#\$1}\geq 1000000000\&];$ $m=\text{Table}\left[\sqrt{v},\{v,\text{list}\}\right];$ $\text{Table}\left[i^2,\{i,\text{Select}[m,\text{IntegerQ}[\text{\$\#\$1}]\&]\}\right]$ $1026753849,1042385796,1098524736,1237069584,1248703569,1278563049,1285437609,1382054976,1436789025,1503267984,1532487609, \\ 1547320896,1643897025,1827049536,1927385604,1937408256,2076351489,2081549376,2170348569,2386517904,2431870596,2435718609, \\ 2571098436,2913408576,3015986724,3074258916,3082914576,3089247561,3094251876,3195867024,3285697041,3412078569,3416987025, \\ 3428570916,3528716409,3719048256,3791480625,3827401956,3928657041,3964087521,3975428601,3985270641,4307821956,4308215769, \\ 4369871025,4392508176,4580176329,4728350169,4730825961,4832057169,5102673489,5273809641,5739426081,5783146209,5803697124, \\ 5982403716,6095237184,6154873209,6457890321,6471398025,6597013284,6714983025,7042398561,7165283904,7285134609,7351862049,\\ 7362154809,7408561329,7680594321,7854036129,7935068241,7946831025,7984316025,8014367529,8125940736,8127563409,8135679204,\\ 8326197504,8391476025,8503421796,8967143025,9054283716,9351276804,9560732841,9614783025,9761835204,9814072356$

SQL

Create a table t with a column f containing the integers 0 to 9 and execute the following query

select min (t1.f + 10 * t2.f + 10^2 * t3.f + 10^3 * t4.f + 10^4 * t5.f + 10^5 * t6.f + 10^6 * t7.f + 10^7 * t8.f + 10^8 * t9.f + 10^9 * t10.f)

from t t1, t t2, t t3, t t4, t t5, t t6, t t7, t t8, t t9, t t10

where t1.f not in (t2.f, t3.f, t4.f, t5.f, t6.f, t7.f, t8.f, t9.f, t10.f)

and t2.f not in (t3.f, t4.f, t5.f, t6.f, t7.f, t8.f, t9.f, t10.f)

and t3.f not in (t4.f, t5.f, t6.f, t7.f, t8.f, t9.f, t10.f)

and t4.f not in (t5.f, t6.f, t7.f, t8.f, t9.f, t10.f)

and t5.f not in (t6.f, t7.f, t8.f, t9.f, t10.f)

and t6.f not in (t7.f, t8.f, t9.f, t10.f)

and t7.f not in (t8.f, t9.f, t10.f)

and t8.f not in (t9.f, t10.f)

and t9.f not in (t10.f)

and sqrt (t1.f + 10 * t2.f + 10^2 * t3.f + 10^3 * t4.f + 10^4 * t5.f + 10^5 * t6.f + 10^6 * t7.f + 10^7 * t8.f + 10^8 * t9.f + 10^9 * t10.f)

= round (sqrt (t1.f + 10 * t2.f + 10^2 * t3.f + 10^3 * t4.f + 10^4 * t5.f + 10^5 * t6.f + 10^6 * t7.f + 10^7 * t8.f + 10^8 * t9.f + 10^9 * t10.f) )