© Friederike Hofmann, MATH+
Author: Lukas Protz (MATH+)
Challenge
Annika is a diligent elf who loves to make her fellow elves happy. This holiday season, she decides to bake a variety of delicious cookies for all her friends – including Santa! To create a wide range of cookie types, she has gathered:
-
- 2 different cookie cutters: A Santa and an elf shape
- ingredients for 2 types of dough: vanilla and chocolate
- 3 flavors of icing: vanilla, chocolate and lime
- 3 types of sprinkles: red hearts, silver pearls and ordinary sprinkles
Given these options, Annika can create exactly 36 unique types of cookies, where each cookie type is defined by a unique combination of cutter, dough, icing, and sprinkles. Two cookies are considered the same type if they use the same cutter, dough, icing, and sprinkles.
To ensure that each type tastes delicious, Annika bakes one of each. Once they are finished, she places all 36 cookies into a bowl.
Annika loves the game SET, and the cookies remind her of it. Now she wonders how many cookies she needs to pick from the bowl to guarantee a SET among the chosen cookies.
A SET of cookies is a collection of three cookies, such that for each category (cutter, dough, …) either all cookies share the same feature or each of the cookies has a different feature. If we represent a cookie via its features (cutter, dough, icing, sprinkles), then for example the cookies
(Santa, vanilla, vanilla, hearts),
(Santa, vanilla, chocolate, pearls),
(Santa, vanilla, lime, ordinary)
form a SET. However, the following three cookies do not form a SET:
(elf, vanilla, vanilla, hearts),
(Santa, vanilla, chocolate, pearls),
(Santa, vanilla, lime, ordinary),
This is, because the first and the second cookie have a different feature in the cutter category, whereas the second and third cookie have the same feature in the cutter category.
Of course, Annika could just pick three of the cookies deliberately to obtain a SET, so to make everything more interesting she picks the cookies at random and without looking. What is the minimum number of cookies Annika needs to select to guarantee that there is at least one SET among the chosen cookies?
Possible Answers
- 14 or more
- 13
- 12
- 11
- 10
- 9
- 8
- 7
- 6
- 5 or less