The following solution was offered by Trevor of Riccarton High School, Christchruch, New Zealand. Other correct solutions were received from Andrei (School 205, Bucharest), Mary (Birchwood Community High School), Rachel and Sebastian (Hethersett High School) and Claire.

Well first, I found all three digit numbers which are square numbers...

$10^2 = 100$
$11^2 = 121$
$12^2 = 144$
$13^2 = 169$
$14^2 = 196$
$15^2 = 225$
$16^2 = 256$
$17^2 = 289$
$18^2 = 324$
$19^2 = 361$
$20^2 = 400$
$21^2 = 441$
$22^2 = 484$
$23^2 = 529$
$24^2 = 576$
$25^2 = 625$
$26^2 = 676$
$27^2 = 729$
$28^2 = 784$
$29^2 = 841$
$30^2 = 900$
$31^2 = 961$

...then I searched for pairs of square numbers that when one is reversed, it will be the same as the second, but not including paladromic numbers...

pairs: $144$, $4414

$169$, $961$

...then the final clue is that Charle's car registration number is a four digit number which is also a square number
as well, and is formed by repeating the last digit of the house number, and using the numbers which I have picked...

$1444$, $4411$
$1699$, $9611$

...I can pick a square number from here, if there is one......$1444$ is a square of $38$! So that means that $144$ is the
house number!