Preliminary 1997-1998

Math A-lympiad: Preliminary 1997-1998

Scratch cards

Scratch cards are popular. They are appealing: just by scratching open the right box, you scratch yourself into a millionaire. By choosing what you scratch you can believe you are influencing your own luck.

There are different scratch systems. In spring the Dutch Railways had a scratch card campaign. When you bought a cup of coffee you were given a scratch card with three squares. You had to scratch open two. If you had two flags with an S on them then you won a free return trip with Stena-Line from Hook of Holland to Harwich. If you scratched two cups, then you got a free cup of coffee. If all three had been scratched open the card was void.
With other scratch systems you can win more money with each square that you open, until you scratch a square with a cross. Then you lose it all. This of course plays on the greed of the person with the card.
This assignment is about this last system.

Scratch it yourself!

The DIY store DHZ will shortly have its 25th birthday. To celebrate this jubilee and to pull in extra customers, the owner decided on the Scratch it yourself! scratch card campaign.
For every f 100 spent the customer was given a scratch card with 10 squares that could be scratched open. By spending f190 you would thus get one card. If you spent f 205 you would get two.
Every card contained five squares with the DHZ logo. These were distributed randomly over the 10 squares.
The following rules of play were printed on the card:

Rules

With one open square showing the DHZ logo you win f 7.50.
With two open squares showing the DHZ logo you win f 15.00.
With every subsequent square showing DHZ the amount you have already won, doubles.
If you open up a cross, you lose everything.

The owner felt that this campaign was going to cost him dear.
One week before starting he did a trial among his staff. He had his employees scratch open 200 cards just as if they were customers. He did not actually pay out the prize money of course.
He got the following results:
52 of the 200 cards had won f 7.50. Fourteen people had opened two logos and won f 15. Four cards had won f 30, one had f 60 and one had f 120. He was shocked by the total prize money for these 200 cards.

Exercise
Analyse this campaign. Take the following points into consideration:

The campaign speculates on the greed and gambling lust of the customer. How is that reflected in the outcome of the trial?
After the trial a number of employees said that they would expect a lot of hassle and moaning from customers. A customer who first opened up a cross and then three logos would then complain that he was entitled to a prize of f 30. Where do you think this fear of the staff comes from?

The owner wanted a different campaign. It had to be less expensive. Nevertheless, the customer had to be able to win back what he spent. The campaign had to be of interest and draw in customers. In addition it should not generate any hassle. He wanted to distribute 10 000 cards. The entire campaign was not to cost him more than 2% of the turnover involved.

Exercise
Devise a different (better, nicer, more sensational) scratch card campaign for the owner. The campaign must satisfy the conditions of the owner:

Main prize at least ƒ100.
Total prizes not more than 2% of turnover.

The owner was open to alternatives. You can thus set up a completely different system.
Show that your design satisfies the requirements of the owner. Also specify why you think your system is better.