Doubling Pennies Etc.

  I researched the history of chess and one of the legends on its origin made me think of the doubled penny problem we often pose students in class.

The problem is the one where you ask a student if they want $1,000,000 or a penny doubled each day for a month.  After exploring the math, they discover the second is the better option.

The story I read said a wise man invented the game of chess to show the king how important each person in his kingdom was.  He enjoyed the game so much, he required all his subjects to learn the job and as a reward, he offered the wise man gold and silver but the man turned it down.  Instead, he requested one grain of wheat on the first square, two on the second, three on the third etc until all 64 squares had been used.  This would have given the wise man quite a lot of grain.  As a matter of fact, the story says the servants scrambled to find enough grain to do this.

Another place this particular doubling problem is found is in bacterial growth.  It is often expressed as starting with one that splits into two, each hour or day and you are asked to find the final numbers at the end of a specific period of time.  Its the same type of problem as the previous two.


All three problems fall under the general term of a doubling problem or exponential growth which is usually applied to how long it takes for a certain population to double.  One way to explain this to students is that it takes over one million microorganisms per milliliter to make water cloudy or contaminated so if it starts with one organism, it may not take long for the bacteria to split into one million organism.

So harmful bacteria grows at the same rate as good bacteria. If one harmful organism enters a pot of stew and splits every hour, the population would have grown to more than eight million which is enough to make someone sick.  This is one reason why you should not leave a pot of food in your trunk on a hot day.

I've seen the bacterial growth problems but I've never seen them applied to harmful bacteria in food which is a real life application of math and explains why there is an occasional outbreak of food poisoning at community potlucks.  It gives students a perspective on why you might want to care about bacterial growth.

So the next time I teach this topic, I'm going to include the example on harmful bacteria which people might find more interesting.  Let me know what you think, I'd love to hear.  Have a great day.