benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Let n be a given positive integer.

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let nn be a given positive integer. A restaurant offers a choice of nn starters, nn main dishes, nn desserts and nn wines. A merry company dines at the restaurant, with each guest choosing a starter, a main dish, a dessert and a wine. No two people place exactly the same order. It turns out that there is no collection of nn guests such that their orders coincide in three of these aspects, but in the fourth one they all differ. (For example, there are no nn people that order exactly the same three courses of food, but nn different wines.) What is the maximal number of guests?
Plain-text mathematical notation (without MathML)
Let n be a given positive integer. 
A restaurant offers a choice of n starters, n main dishes, n desserts and n wines. 
A merry company dines at the restaurant, with each guest choosing a starter, a main dish, a dessert and a wine.  
No two people place exactly the same order. 
It turns out that there is no collection of n guests such that their orders coincide in three of these aspects, 
but in the fourth one they all differ. (For example, there are no n people that order exactly the same three courses of food, but n different wines.) What is the maximal number of guests?
Original LaTeX notation
Let $n$ be a given positive integer. 
A restaurant offers a choice of $n$ starters, $n$ main dishes, $n$ desserts and $n$ wines. 
A merry company dines at the restaurant, with each guest choosing a starter, a main dish, a dessert and a wine.  
No two people place exactly the same order. 
It turns out that there is no collection of $n$ guests such that their orders coincide in three of these aspects, 
but in the fourth one they all differ. (For example, there are no $n$ people that order exactly the same three courses of food, but $n$ different wines.) What is the maximal number of guests?

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
9450688d-8b96-5e77-9e82-a7d5b896e8a0
task_key
test--9450688d-8b96-5e77-9e82-a7d5b896e8a0
task_revision_id
1
config / split
/ test

Official source

initial import