These solutions, which use quite different methods, are by Ang Zhi Ping, age 16, River Valley High School, Singapore and Pierce Geoghegan 17, Tarbert Comprehensive, Ireland. Good work Ang Zhi and Pierce!

Dividing these numbers by 11:

209 = 11x19;

308 = 11x28;

407 =11x37;

506 = 11x46;

605 = 11x55;

704 = 11x64;

803 = 11x73 and

902= 11x82.

The only number in this list which is itself a square number (11 times a square number) is 704 so the only number satisfying the required conditions is 7744.

As observed, 700+4=704=11x64, thus, satisfying the conditions that 100a+b is a multiple of 11, and (100a+b)/ 11 is another square (64=8 ²). The number is 7744, equal to 88 ² .

This solution comes from Pierce Geoghegan: