Weihnachtsmann rot

Challenge from 16. December

Head Protecting Hats

In an earlier version of the challenge, the percentage p was wrongly definined.

We are very sorry for the inconvenience!

Author: Paul Erchinger

Project: MATH+ School Activities


It's Christmas time. In Elf Town, the anticipation of the coming festive season is almost unstoppable. There is only one worry: since November, there has been a dramatic increase in cases of Halloween Frightening Fleas. This is a particularly sneaky species of fleas that settles on the heads of its hosts and infects them with the spook. Elves affected by the spook give others a proverbial heart attack (without warning and at random times). To ensure the well-being of the population, elf mayor Alva and her team of experts must develop a strategy to contain this threat to the spirit of Christmas.

A flea infection happens as follows: if a flea-infested person and a “healthy” person meet, the cunning fleas will definitely hop over to the other person. Afterwards both persons will be infected. It is courtesy that every inhabitant of Elf Town wears a hat (as soon as she leaves the house). Although it is very fashionable, this normal hat does not protect the elves from a flea infection: the fleas simply crawl out of the hat of an infected elf and jump onto the hat of another elf. There, they crawl underneath the hat to reach her head.

Therefore, researchers of Elf Town have developed the novel hat IBeanie, which drastically reduces the risk of infection:

  • In 50 % of the cases, the IBeanie prevents the fleas from crawling out of the hat.

  • In 95 % of the cases, the fleas are not able to get through the hat (to the head) from the outside.

Hence, if two elves wearing IBeanies meet, one of whom is infected with fleas, the risk of infection for the other one is only 2.5 %.

From experience, we know that some of the inhabitants of Elf Town will refuse to exchange their stylish normal hat for the somewhat clumsier new IBeanie. However, if two people meet, only one of whom is wearing the IBeanie, the risk of infection is correspondingly higher.

Alva wants that the risk of infection for when any healthy person meets any infected person is at most 13 %.

Let p be the percentage of Elf Town residents who do wear the IBeanie. What is the smallest integer p such that Alva's condition is satisfied? More precisely, what is the digit sum of that p?

Artwork: Frauke Jansen

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Possible answers:

  1. The digit sum of p is 1.

  2. The digit sum of p is 3.

  3. The digit sum of p is 5.

  4. The digit sum of p is 7.

  5. The digit sum of p is 9.

  6. The digit sum of p is 11.

  7. The digit sum of p is 13.

  8. The digit sum of p is 15.

  9. The digit sum of p is 17.

  10. There exists no such p.

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