Answers that I still find surprising

It is always a little nice when I still find some of the answers I get for questions I give to students surprising.  It is not that the answers are unexpected, it is more that sometimes I get excited by the solutions.

A local supermarket chain on and off for the last few years has been running a bit of a promotion.  The promotion involves a series of 108 animal cards.  For every $20 spent at their store you get a pack of 4 animal cards.  I have a three year old daughter who loves these cards. She loves collecting them and seeing which ones we have missing.  I know that over the course of the promotion we have collected a lot of cards and have got a lot of doubles, it got me wondering how many cards I would need to collect (doubles and all) to collect the full set of 108.

I decided to try test this out with a series of students who were making the transition from year 7 to year 8.  We used Excel to simulate this situation.  The images to the right show some of the output from these simulations.

The image in the top left shows the approximate situation after 200 cards, most of them have been collected as can be seen in green but there are also about 20 gaps there.  The 200 cards were simulated various times with sometimes more gaps.

and sometimes less gaps, but the 20 represents the average number of cards missing.  At 400 cards the average missing drops to about 2, a substantial gain in cards.  However at this point you have spent $2000 dollars at their store.  In running the multiple simulations of the 400 cards there was only one or two occasions out of the dozens of times tested that I managed to get a full set of 108 out of the 400 cards.

Raising this again to 600 cards there is still a gap.  about a third of the time there was a full set cards, but in two thirds of cases there was an average of one card missing, it was only when taking the number of cards to 800 that we began to consistently got a full set of cards.  At 800 cards you have spent $4000 in their stores.

Having to spend $4000 to in a sense guarantee a full set of cards that probably cost them less than $5 to produce is not necessarily something that sits well with me.  Part of the money generated is being used to support a few zoo's, so it is not so bad.   In reality people will swap cards with other people to fill the gaps, and they are likely to spend this money any way, they are not doing more shopping just to get cards.  However if the parents of these children used to shop at another supermarket chain, then the introduction of these cards. if only for a short time, has no doubt swung quite a bit of business their way.  These were the sort of discussions that were so valuable in conducting this task with the class.