This is a variation on a trick I've covered previously for multiplying numbers just above 100. But since the numbers are the same, it lets us simplify a couple of the steps.
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Take whatever amount the number is above 100 and square that. That will be the final two digits of your square (assuming you're sticking to 109 or below). Then take that same number and double it to get the next two digits of the square. Finally, since the number we're squaring is just above 100, its square must be just above 100², which is 10,000. So write down a 1 as the first digit, and you're done!
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