Competition problem — Combinatorics

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Combinatorics Difficulty 5.6 AIME, harder Prove it United States

Problem:

Eight friends, Aerith, Bob, Chebyshev, Descartes, Euler, Fermat, Gauss, and Hilbert, bought tickets for adjacent seats at the opera. However when they arrived they mixed up their seats:
- Bob sat in his assigned seat,
- Chebyshev sat two seats to the right of Gauss' assigned seat,
- Descartes sat one seat to the left of Fermat's assigned seat,
- Euler sat four seats to the left of Hilbert's assigned seat,
- Fermat sat five seats to the right of Descartes' assigned seat,
- Gauss sat one to the right of Euler's assigned seat,
- Hilbert sat three seats to the left of Aerith's assigned seat.
In whose seat did Aerith sit?

Solution

Solution:

Number the seats $1$ through $8$ and let $a, \ldots, h$ be the seat assignments. Let $A$ be the seat occupied by Aerith. As each seat is assigned to exactly one person we must have $a+\cdots+h=1+\cdots+8$ . As each seat is occupied by exactly one person we must have
$\begin{aligned} 1+\cdots+8 & =A+b+(g+2)+(f-1)+(h-4)+(d+5)+(e+1)+(a-3) \\ & =A+(a+b+d+e+f+g+h) . \end{aligned}$
Thus $A=c$ , so Aerith occupies Chebyshev's seat.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.