Weihnachtsmann rot

Challenge from 7. December

Tricky Testing Task

For this challenge, you will have a time bonus of 48 hours.

Author: Paul Erchinger

Project: MATH+ School Activities

Challenge:

Oh deer! It's Christmas flu season. Normally, this does not pose a problem for the Christmas preparations, because Christmas elves are immune to this virus. But this year, a particularly insidious Easter mutation seems to circulate. With only four weeks left until Christmas, several of Santa's little helpers are infected, having grown long ears and whiskers. Hence, the Christmas crisis management team meets and decides that the elves need to be tested thoroughly. The plan is to quickly find sick elves, before even more of them become infected.

Of course, the test has to be carried out at the Christmas headquarters at the North Pole too. But testing capacities in the Arctic are quite scarce: due to massive cracks in the ice floe landscape, the sledge, which usually delivers all of the supplies, cannot reach the headquarters. Hence, for Santa and his 14 elves, there are only ten CE tests1 left. Fortunately, they were delivered by mail some time ago.

Santa is at a loss: how can he test the entire staff of 15 people without the sufficient capacity? Furthermore, at this year's Halloween party, all elves could have infected one another. Fortunately, elf Annelie has an idea, “Do you remember our seating arrangements for Halloween? We sat down to dine at five tables. In order to not get bored so easily, we changed seats after the main course. Thus, no one sat at a table twice with the same elves. I should still have the list somewhere...” After a short rummage, she pulls the list out of a box containing fluffy pink Easter Bunny overalls:

“We just take two samples from each of us and mix the samples of those who shared the same table. This way, we only need to do five tests for the main course and five for the dessert, which amounts to only ten tests! Then, we simply identify those elves as sick who were in groups that tested positive in both test runs. Those elves need to isolated from the others.”

A lively discussion begins:

Boris states,
“With this strategy, we will never find all the infected elves...”

Annelie responds,
“On the contrary! We will always find all the infected ones.”

Carmen reckons,
“That's right! But we also get false positive test results. That means: elves who are not infected at all could still be tested positive. This time of the year, we cannot afford them to be absent from work...”

Jakob replies,
“But, since we never shared both tables with the same elves, there will be no false positive test results.”

Mara has a different opinion,
“Unfortunately, your statement is not correct. However, if we knew that at most two people were infected, it would be true.”

Lorenz believes,
“No! Even then, some elves could get a false positive test result.”

Esmeralda has another idea,
„We just have to divide the groups in a different way:

Moreover, if we knew that at most two of us were infected, we could always find out unambiguously who it was.”

Santa is as clueless as before: who is right? After all, he has to organise the whole testing...

1 It is known that elf cells are normally strongly attracted to Christmas symbols. Therefore, the CE test involves holding both a Christmas tree ball and an Easter egg in the test sample. If the cells react more strongly to the Easter egg, they are infected by the Easter mutated virus. Unfortunately, a fresh Easter egg is needed for each sample, which is why the tests are very resource-consuming and therefore need to be rationed.

Illustration: Frauke Jansen

PDF download

Possible answers:

  1. Only Boris.

  2. Only Annelie and Carmen.

  3. Only Annelie and Jakob.

  4. Only Annelie, Carmen, and Mara.

  5. Only Annelie, Carmen, and Lorenz.

  6. Only Boris and Esmeralda.

  7. Only Annelie, Carmen, and Esmeralda.

  8. Only Annelie, Jakob, and Esmeralda.

  9. Only Annelie, Carmen, Mara, and Esmeralda.

  10. Only Annelie, Carmen, Lorenz, and Esmeralda.

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