In a chess tournament, there are four groups, each containing an equal number of players. Each player plays 1. against every other player belonging to one's own group exactly once; 2. against each player belonging to one of the remaining three groups exactly twice; 3. against each player belonging to one of the remaining two groups exactly three times; and 4. against each player belonging to the remaining group exactly four times. If there are more than 1000 matches being played in the tournament, the minimum possible number of players in each group is

Answer: 8. Solution: 4 teams: A, B, C, D, each with n players. We count matches from each team's perspective, being careful not to double-count. Counting matches involving each team:, Team, Opponent, Frequency, Matches, A, A (internal), once, ^nC_2, A, B, twice, 2n^2, A, C, thrice, 3n^2, A, D, 4 times, 4n^2, B, B (internal), once, ^nC_2, B, C, 4 times, 4n^2, B, D, thrice, 3n^2, B, A, twice, -- (counted in A), C, C (internal), once, ^nC_2, C, D, twice, 2n^2, C, A, B, thrice / 4x, -- (counted above), D, D (internal), once, ^nC_2, D, A, B, C, all, -- (counted above), Internal matches (4 teams): 4 x ^nC_2 = 4 x (n(n-1))/2 = 2n(n-1) Cross-team matches (unique pairs, summing frequencies):, Pair, Frequency, Matches, A-B, 2, 2n^2, A-C, 3, 3n^2, A-D, 4, 4n^2, B-C, 4, 4n^2, B-D, 3, 3n^2, C-D, 2, 2n^2, Total, 18n^2, Total Matches: = 4(^nC_2) + 18n^2 > 1000 = 2n(n-1) + 18n^2 > 1000 = 20n^2 - 2n > 1000 Testing n = 7: 20(49) - 14 = 980 - 14 = 966 1000 Least integral value of n = 8.

Modern Math

Permutation & Combination

Hard

In a chess tournament, there are four groups, each containing an equal number of players. Each player plays

against every other player belonging to one's own group exactly once;

against each player belonging to one of the remaining three groups exactly twice;

against each player belonging to one of the remaining two groups exactly three times; and

against each player belonging to the remaining group exactly four times.

If there are more than 1000 matches being played in the tournament, the minimum possible number of players in each group is

Entered answer:

✅ Correct Answer: 8

4 teams: A, B, C, D, each with $n$ players.

We count matches from each team's perspective, being careful not to double-count.

Counting matches involving each team:

Team	Opponent	Frequency	Matches
A	A (internal)	once	${}^{n}C_{2}$
A	B	twice	$2n^2$
A	C	thrice	$3n^2$
A	D	4 times	$4n^2$
B	B (internal)	once	${}^{n}C_{2}$
B	C	4 times	$4n^2$
B	D	thrice	$3n^2$
B	A	twice	-- (counted in A)
C	C (internal)	once	${}^{n}C_{2}$
C	D	twice	$2n^2$
C	A, B	thrice / 4x	-- (counted above)
D	D (internal)	once	${}^{n}C_{2}$
D	A, B, C	all	-- (counted above)

Internal matches (4 teams): $4 \times {}^{n}C_{2} = 4 \times \dfrac{n(n-1)}{2} = 2n(n-1)$

Cross-team matches (unique pairs, summing frequencies):

Pair	Frequency	Matches
A-B	2	$2n^2$
A-C	3	$3n^2$
A-D	4	$4n^2$
B-C	4	$4n^2$
B-D	3	$3n^2$
C-D	2	$2n^2$
Total		$18n^2$

Total Matches:

$= 4({}^{n}C_{2}) + 18n^2 > 1000$

$= 2n(n-1) + 18n^2 > 1000$

$= 20n^2 - 2n > 1000$

Testing $n = 7$ : $20(49) - 14 = 980 - 14 = 966 < 1000$

Testing $n = 8$ : $20(64) - 16 = 1280 - 16 = 1264 > 1000$

Least integral value of $n$ = 8.

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