PSYLLOGISM: Alarm Clock

Alarm Clock

BEEP BEEP BEEP BEEP BEEP BEEP BEEP BEEP

My alarm clock starts going off at 5:33 AM, as it does almost every weekday morning. I hit the snooze button.

BEEP BEEP BEEP BEEP BEEP BEEP BEEP BEEP

Nine minutes later, at 5:42 AM, my alarm clock wakes me up again. I've always wondered why nine minutes seems to be the standard with many models of alarm clocks?

BEEP BEEP BEEP BEEP BEEP BEEP BEEP BEEP

It's 5:51 AM. "Hmm, that's interesting," my tired mind exclaims. "The digits for each of the 'minute' part of those times adds up to six. 3+3=6, 4+2=6, 5+1=6…Now why would that be?"

I roll over in bed, waking up to this thought process: "Well, is that true for all two digit numbers? Let's take a number made up of two digits, $a$ and $b$ . Then that number is equal to $10a + b$ . Then we add 9 to it, so we get $10a + b + 9$ . Well, 9 is just 10 - 1, so we have $10a + b + 10 - 1 = 10(a+1) + (b-1)$ , which, if $a < 9$ and $b > 0$ , is another two digit number, with digits $a+1$ and $b-1$ . And look, $(a-1)+(b+1) = a + b$ , so the sum of the digits is the same! Eureka! I have found it! That's why the digits for the minute parts all add up to six!