I am currently attending a conference that is housing everybody in a rather large hotel. When I checked in last night I was given room 3025.

My first thought when I looked at the number was “Ah, this is a perfect square”. So as I carried my bags to the room, I mentally set about working out the square root of my room number.

Being lazy, I did this the easiest way I could think of: I multiplied by four to find a number with zeros at the end. This gave me 12100. Which is clearly 121 × 100, or 11^{2} × 10^{2}, or 110^{2}.

Having multiplied by four to get the square of 110, I now needed to divide 110 by two. So by the time I arrived at my room, I knew that the square root of 3025 is 55.

But how did I know, in that first moment at the registration desk, that 3025 is a perfect square?

I remember that I had immediately been struck by seeing “30” followed by “25”. This suggests that I may have subconsciously recognized the visual pattern 2500 + 500 + 25 as an instance of (a^{2})100 + (2a)10 + a^{2}, or (10a + a)^{2}.

If this is true, it would mean that my conscious and subconscious minds had gone in completely different directions. And it would also mean that my subconscious mind had been a whole lot quicker than my conscious mind.

Alas, if only my conscious mind could be as quick as my subconscious mind.

Here is an interesting method to calculate the square of 10n+5. Write down the result of n(n+1), then append 25 to the right side of it. Actually I mean (10n+5)^2 = n(n+1)*100 + 25. Now you must be able to tell the square root of anything like this, such as 3025, 4225, and 7225.