Tuesday, February 14th, 2012...11:11 am

Sum kiss!

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Suppose this Valentine’s Day your sweetheart asks what you’d like as a gift. You innocently request, “How about you give me one kiss right now, twice as many in an hour and twice that the next hour?” Just one simple kiss right now, 2 kisses in an hour and 4 kisses in the next hour!

It’s about 11 o’clock in the morning at the time of this post. How many kisses will you have by 10 PM, which is 11 hours away? How many kisses would you be getting at 10 PM? You start with 1 kiss, after one hour you receive 2^1, after two 2^2 and so forth. So, after 11 hours you would receive 2^{11} = 2048 kisses which would take just over 34 minutes if you received a peck on the cheek at 1 pps (pecks per second). Just think, if your love had asked before 10 AM, then by 10 PM, the kissing at 1 pps would happen continually for those final hours!

Now, how many total kisses would you receive over that time? Recall

 sum_{k=0}^n 2^k = 2^{k+1}-1

So, after 11 hours, you get 2^{12} - 1 = 4095 kisses. That’s sum kissing!



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