Five players – A, B, C, D and E – took part in a round-robin chess tournament in which every player played every other player exactly once. Before the tournament their ratings were:
| Player | A | B | C | D | E |
|---|---|---|---|---|---|
| Pre-tournament rating | 2400 | 2350 | 2300 | 2250 | 2200 |
After each game, rating points were transferred between the two players as follows. Let g be the difference between the two players' pre-tournament ratings (so g is always a multiple of 50), and let k = g/25.
- If the higher-rated player won, the loser transferred (10 − k) points to the winner.
- If the lower-rated player won, the loser transferred (10 + k) points to the winner.
- If the game was drawn, the higher-rated player transferred k points to the lower-rated player.
No other points were gained or lost. The table below gives each player's record in the tournament (W = wins, D = draws, L = losses).
| Player | W | D | L |
|---|---|---|---|
| A | 3 | 0 | 1 |
| B | 2 | 1 | 1 |
| C | 0 | 1 | 3 |
| D | 1 | 0 | 3 |
| E | 2 | 2 | 0 |
| Player | A | B | C | D | E |
|---|---|---|---|---|---|
| Pre-tournament rating | 2400 | 2350 | 2300 | 2250 | 2200 |
After each game, rating points were transferred between the two players as follows. Let g be the difference between the two players' pre-tournament ratings (so g is always a multiple of 50), and let k = g/25.
- If the higher-rated player won, the loser transferred (10 − k) points to the winner.
- If the lower-rated player won, the loser transferred (10 + k) points to the winner.
- If the game was drawn, the higher-rated player transferred k points to the lower-rated player.
No other points were gained or lost. The table below gives each player's record in the tournament (W = wins, D = draws, L = losses).
| Player | W | D | L |
|---|---|---|---|
| A | 3 | 0 | 1 |
| B | 2 | 1 | 1 |
| C | 0 | 1 | 3 |
| D | 1 | 0 | 3 |
| E | 2 | 2 | 0 |
Question 1
Single correct option
Which players did E defeat?
- A. A and B
- B. A and D
- C. C and D
- D. B and D
Show answer and solution
Answer: (B) A and D
Deducing the results. Total draws = (0 + 1 + 1 + 0 + 2)/2 = 2 games. E drew twice and the only other players with a draw are B and C, so the draws are B–E and C–E. E lost nothing, so E beat A and D.
A lost only once (to E), so A beat B, C and D. C won nothing and drew only with E, so C lost to A, B and D. D's one win must then be over C, so D lost to B. B: beat C and D, lost to A, drew with E (2-1-1 ✓).
| Game | Result | k | Points moved |
|---|---|---|---|
| A–B | A won | 2 | 8 to A |
| A–C | A won | 4 | 6 to A |
| A–D | A won | 6 | 4 to A |
| A–E | E won (upset) | 8 | 18 to E |
| B–C | B won | 2 | 8 to B |
| B–D | B won | 4 | 6 to B |
| B–E | draw | 6 | 6 to E |
| C–D | D won (upset) | 2 | 12 to D |
| C–E | draw | 4 | 4 to E |
| D–E | E won (upset) | 2 | 12 to E |
Final ratings: A 2400, B 2350, C 2270, D 2240, E 2240.
So E defeated A and D and drew with B and C.
A lost only once (to E), so A beat B, C and D. C won nothing and drew only with E, so C lost to A, B and D. D's one win must then be over C, so D lost to B. B: beat C and D, lost to A, drew with E (2-1-1 ✓).
| Game | Result | k | Points moved |
|---|---|---|---|
| A–B | A won | 2 | 8 to A |
| A–C | A won | 4 | 6 to A |
| A–D | A won | 6 | 4 to A |
| A–E | E won (upset) | 8 | 18 to E |
| B–C | B won | 2 | 8 to B |
| B–D | B won | 4 | 6 to B |
| B–E | draw | 6 | 6 to E |
| C–D | D won (upset) | 2 | 12 to D |
| C–E | draw | 4 | 4 to E |
| D–E | E won (upset) | 2 | 12 to E |
Final ratings: A 2400, B 2350, C 2270, D 2240, E 2240.
So E defeated A and D and drew with B and C.
Question 2
Numerical answer
What was C's rating at the end of the tournament?
Show answer and solution
Answer: 2270
C lost to A (higher-rated won, k = 4: C gives 6), lost to B (k = 2: gives 8), lost to D (lower-rated D won, k = 2: C gives 10 + 2 = 12) and drew with E (C is higher-rated, k = 4: C gives 4).
Total loss = 6 + 8 + 12 + 4 = 30, so C ends at 2300 − 30 = 2270.
Deducing the results. Total draws = (0 + 1 + 1 + 0 + 2)/2 = 2 games. E drew twice and the only other players with a draw are B and C, so the draws are B–E and C–E. E lost nothing, so E beat A and D.
A lost only once (to E), so A beat B, C and D. C won nothing and drew only with E, so C lost to A, B and D. D's one win must then be over C, so D lost to B. B: beat C and D, lost to A, drew with E (2-1-1 ✓).
| Game | Result | k | Points moved |
|---|---|---|---|
| A–B | A won | 2 | 8 to A |
| A–C | A won | 4 | 6 to A |
| A–D | A won | 6 | 4 to A |
| A–E | E won (upset) | 8 | 18 to E |
| B–C | B won | 2 | 8 to B |
| B–D | B won | 4 | 6 to B |
| B–E | draw | 6 | 6 to E |
| C–D | D won (upset) | 2 | 12 to D |
| C–E | draw | 4 | 4 to E |
| D–E | E won (upset) | 2 | 12 to E |
Final ratings: A 2400, B 2350, C 2270, D 2240, E 2240.
Total loss = 6 + 8 + 12 + 4 = 30, so C ends at 2300 − 30 = 2270.
Deducing the results. Total draws = (0 + 1 + 1 + 0 + 2)/2 = 2 games. E drew twice and the only other players with a draw are B and C, so the draws are B–E and C–E. E lost nothing, so E beat A and D.
A lost only once (to E), so A beat B, C and D. C won nothing and drew only with E, so C lost to A, B and D. D's one win must then be over C, so D lost to B. B: beat C and D, lost to A, drew with E (2-1-1 ✓).
| Game | Result | k | Points moved |
|---|---|---|---|
| A–B | A won | 2 | 8 to A |
| A–C | A won | 4 | 6 to A |
| A–D | A won | 6 | 4 to A |
| A–E | E won (upset) | 8 | 18 to E |
| B–C | B won | 2 | 8 to B |
| B–D | B won | 4 | 6 to B |
| B–E | draw | 6 | 6 to E |
| C–D | D won (upset) | 2 | 12 to D |
| C–E | draw | 4 | 4 to E |
| D–E | E won (upset) | 2 | 12 to E |
Final ratings: A 2400, B 2350, C 2270, D 2240, E 2240.
Question 3
Numerical answer
What was the largest number of rating points transferred in any single game of the tournament?
Show answer and solution
Answer: 18
The biggest transfers come from upsets across large rating gaps. The upsets were E over A (g = 200, k = 8: 10 + 8 = 18), D over C (k = 2: 12) and E over D (k = 2: 12). No win by a higher-rated player can move more than 10 − 2 = 8 points and no draw more than 8.
Maximum = 18 (E's win over A).
Deducing the results. Total draws = (0 + 1 + 1 + 0 + 2)/2 = 2 games. E drew twice and the only other players with a draw are B and C, so the draws are B–E and C–E. E lost nothing, so E beat A and D.
A lost only once (to E), so A beat B, C and D. C won nothing and drew only with E, so C lost to A, B and D. D's one win must then be over C, so D lost to B. B: beat C and D, lost to A, drew with E (2-1-1 ✓).
| Game | Result | k | Points moved |
|---|---|---|---|
| A–B | A won | 2 | 8 to A |
| A–C | A won | 4 | 6 to A |
| A–D | A won | 6 | 4 to A |
| A–E | E won (upset) | 8 | 18 to E |
| B–C | B won | 2 | 8 to B |
| B–D | B won | 4 | 6 to B |
| B–E | draw | 6 | 6 to E |
| C–D | D won (upset) | 2 | 12 to D |
| C–E | draw | 4 | 4 to E |
| D–E | E won (upset) | 2 | 12 to E |
Final ratings: A 2400, B 2350, C 2270, D 2240, E 2240.
Maximum = 18 (E's win over A).
Deducing the results. Total draws = (0 + 1 + 1 + 0 + 2)/2 = 2 games. E drew twice and the only other players with a draw are B and C, so the draws are B–E and C–E. E lost nothing, so E beat A and D.
A lost only once (to E), so A beat B, C and D. C won nothing and drew only with E, so C lost to A, B and D. D's one win must then be over C, so D lost to B. B: beat C and D, lost to A, drew with E (2-1-1 ✓).
| Game | Result | k | Points moved |
|---|---|---|---|
| A–B | A won | 2 | 8 to A |
| A–C | A won | 4 | 6 to A |
| A–D | A won | 6 | 4 to A |
| A–E | E won (upset) | 8 | 18 to E |
| B–C | B won | 2 | 8 to B |
| B–D | B won | 4 | 6 to B |
| B–E | draw | 6 | 6 to E |
| C–D | D won (upset) | 2 | 12 to D |
| C–E | draw | 4 | 4 to E |
| D–E | E won (upset) | 2 | 12 to E |
Final ratings: A 2400, B 2350, C 2270, D 2240, E 2240.
Question 4
Single correct option
Which of the following players finished the tournament with exactly the same rating they started with?
- A. A only
- B. B only
- C. A and B
- D. B and E
Show answer and solution
Answer: (C) A and B
A: +8 (vs B) + 6 (vs C) + 4 (vs D) − 18 (vs E) = 0.
B: −8 (vs A) + 8 (vs C) + 6 (vs D) − 6 (draw with E) = 0.
C: −30, D: −4 − 6 + 12 − 12 = −10, E: +18 + 6 + 4 + 12 = +40.
So A and B both finish exactly where they started. (Check: net changes 0 + 0 − 30 − 10 + 40 = 0 ✓.)
Deducing the results. Total draws = (0 + 1 + 1 + 0 + 2)/2 = 2 games. E drew twice and the only other players with a draw are B and C, so the draws are B–E and C–E. E lost nothing, so E beat A and D.
A lost only once (to E), so A beat B, C and D. C won nothing and drew only with E, so C lost to A, B and D. D's one win must then be over C, so D lost to B. B: beat C and D, lost to A, drew with E (2-1-1 ✓).
| Game | Result | k | Points moved |
|---|---|---|---|
| A–B | A won | 2 | 8 to A |
| A–C | A won | 4 | 6 to A |
| A–D | A won | 6 | 4 to A |
| A–E | E won (upset) | 8 | 18 to E |
| B–C | B won | 2 | 8 to B |
| B–D | B won | 4 | 6 to B |
| B–E | draw | 6 | 6 to E |
| C–D | D won (upset) | 2 | 12 to D |
| C–E | draw | 4 | 4 to E |
| D–E | E won (upset) | 2 | 12 to E |
Final ratings: A 2400, B 2350, C 2270, D 2240, E 2240.
B: −8 (vs A) + 8 (vs C) + 6 (vs D) − 6 (draw with E) = 0.
C: −30, D: −4 − 6 + 12 − 12 = −10, E: +18 + 6 + 4 + 12 = +40.
So A and B both finish exactly where they started. (Check: net changes 0 + 0 − 30 − 10 + 40 = 0 ✓.)
Deducing the results. Total draws = (0 + 1 + 1 + 0 + 2)/2 = 2 games. E drew twice and the only other players with a draw are B and C, so the draws are B–E and C–E. E lost nothing, so E beat A and D.
A lost only once (to E), so A beat B, C and D. C won nothing and drew only with E, so C lost to A, B and D. D's one win must then be over C, so D lost to B. B: beat C and D, lost to A, drew with E (2-1-1 ✓).
| Game | Result | k | Points moved |
|---|---|---|---|
| A–B | A won | 2 | 8 to A |
| A–C | A won | 4 | 6 to A |
| A–D | A won | 6 | 4 to A |
| A–E | E won (upset) | 8 | 18 to E |
| B–C | B won | 2 | 8 to B |
| B–D | B won | 4 | 6 to B |
| B–E | draw | 6 | 6 to E |
| C–D | D won (upset) | 2 | 12 to D |
| C–E | draw | 4 | 4 to E |
| D–E | E won (upset) | 2 | 12 to E |
Final ratings: A 2400, B 2350, C 2270, D 2240, E 2240.
Question 5
Single correct option
If E had won the game against B instead of drawing it (all other results unchanged), what would E's final rating have been?
- A. 2246
- B. 2250
- C. 2256
- D. 2234
Show answer and solution
Answer: (B) 2250
For B–E, g = 150 so k = 6. In the actual draw E gained k = 6. Had E (the lower-rated player) won, E would have gained 10 + 6 = 16.
E's actual final rating = 2200 + 18 + 6 + 4 + 12 = 2240. Replacing the +6 by +16: 2240 − 6 + 16 = 2250.
(2256 adds 16 without removing the 6; 2246 uses the higher-rated winner's 10 − k.)
Deducing the results. Total draws = (0 + 1 + 1 + 0 + 2)/2 = 2 games. E drew twice and the only other players with a draw are B and C, so the draws are B–E and C–E. E lost nothing, so E beat A and D.
A lost only once (to E), so A beat B, C and D. C won nothing and drew only with E, so C lost to A, B and D. D's one win must then be over C, so D lost to B. B: beat C and D, lost to A, drew with E (2-1-1 ✓).
| Game | Result | k | Points moved |
|---|---|---|---|
| A–B | A won | 2 | 8 to A |
| A–C | A won | 4 | 6 to A |
| A–D | A won | 6 | 4 to A |
| A–E | E won (upset) | 8 | 18 to E |
| B–C | B won | 2 | 8 to B |
| B–D | B won | 4 | 6 to B |
| B–E | draw | 6 | 6 to E |
| C–D | D won (upset) | 2 | 12 to D |
| C–E | draw | 4 | 4 to E |
| D–E | E won (upset) | 2 | 12 to E |
Final ratings: A 2400, B 2350, C 2270, D 2240, E 2240.
E's actual final rating = 2200 + 18 + 6 + 4 + 12 = 2240. Replacing the +6 by +16: 2240 − 6 + 16 = 2250.
(2256 adds 16 without removing the 6; 2246 uses the higher-rated winner's 10 − k.)
Deducing the results. Total draws = (0 + 1 + 1 + 0 + 2)/2 = 2 games. E drew twice and the only other players with a draw are B and C, so the draws are B–E and C–E. E lost nothing, so E beat A and D.
A lost only once (to E), so A beat B, C and D. C won nothing and drew only with E, so C lost to A, B and D. D's one win must then be over C, so D lost to B. B: beat C and D, lost to A, drew with E (2-1-1 ✓).
| Game | Result | k | Points moved |
|---|---|---|---|
| A–B | A won | 2 | 8 to A |
| A–C | A won | 4 | 6 to A |
| A–D | A won | 6 | 4 to A |
| A–E | E won (upset) | 8 | 18 to E |
| B–C | B won | 2 | 8 to B |
| B–D | B won | 4 | 6 to B |
| B–E | draw | 6 | 6 to E |
| C–D | D won (upset) | 2 | 12 to D |
| C–E | draw | 4 | 4 to E |
| D–E | E won (upset) | 2 | 12 to E |
Final ratings: A 2400, B 2350, C 2270, D 2240, E 2240.
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