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One Wasn't Square

Age 7 to 11 Challenge Level:

Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children: Mona, Bob and Jamie.

3 children

"Now", she said, "Those three numbers add to a special kind of number. What is it?"

Michael put his hand up.

"It's a square number", he answered.

"Correct", smiled Mrs Morgan.

"Oh!" exclaimed Mona, "The two numbers I can see also add to a square!"

"And me!" called out Bob, "The two numbers I can see add to a square too!"

"Oh dear", said Jamie disappointedly, "the two numbers I can see don't add to a square! It's either $5$ too little or $6$ too big!"

What numbers did the three children have on their backs?