benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Let n be a positive integer. A [i]Nordic[/i] square is an n×n board containing all the integers from 1 to n² so that each cell contains exactly one number. Two different cells are …

Problem

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

problem

Let nn be a positive integer. A [i]Nordic[/i] square is an n×nn \times n board containing all the integers from 11 to n2n^2 so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that: (i) the first cell in the sequence is a valley, (ii) each subsequent cell in the sequence is adjacent to the previous cell, and (iii) the numbers written in the cells in the sequence are in increasing order. Find, as a function of nn, the smallest possible total number of uphill paths in a Nordic square. Author: Nikola Petrovi?
Plain-text mathematical notation (without MathML)
Let n be a positive integer. A [i]Nordic[/i] square is an n×n board containing all the integers from 1 to n² so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that:

(i) the first cell in the sequence is a valley,

(ii) each subsequent cell in the sequence is adjacent to the previous cell, and

(iii) the numbers written in the cells in the sequence are in increasing order.

Find, as a function of n, the smallest possible total number of uphill paths in a Nordic square.

Author: Nikola Petrovi?
Original LaTeX notation
Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a [i]valley[/i]. An [i]uphill path[/i] is a sequence of one or more cells such that:

(i) the first cell in the sequence is a valley,

(ii) each subsequent cell in the sequence is adjacent to the previous cell, and

(iii) the numbers written in the cells in the sequence are in increasing order.

Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square.

Author: Nikola Petrovi?

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
8a8cabee-8a5a-55b1-9411-b0e2932cf581
task_key
test--8a8cabee-8a5a-55b1-9411-b0e2932cf581
task_revision_id
1
config / split
/ test

Official source

initial import