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Oliveboard Question of the day

At one stage of IPL, DeVilliers, Milller, Maxwell and Gayle were among the top four batsmen in the tournament run chart. The aggregates of their runs were the perfect squares of four consecutive numbers. In the encounter between Bangalore and Punjab, Gayle and DeVilliers were the star performers for Bangalore. Gayle’s score was more than 100 which is a perfect square of a number and he jumped from the 4th to the 2nd position. DeVilliers got 80 and retained his top spot. Maxwell and Miller, who play for Punjab, both scored less than 50. Coincidentally, at the end of the match, their aggregates were still perfect squares of numbers.

What was the total number of runs scored by these four batsmen in the match?
(a) 252
(b) 280
(c) 296
(d) 269

Solution posted below. Click on the plus sign to view the solution.

Answer: Option (C) It is clear from the given statements that prior to the match, DeVilliers was leading the chart and Gayle was at the 4th position. It is also given that DeVilliers scored 80 and his new aggregate is also a perfect square. So we need to express 80 as the difference of two squares.

a2 – b2 = 80 or (a+b)(a-b)=80

For (a+b) and (a-b) both to be whole numbers, either both are even or both are odd. Since 80 is an even number, both its factors cannot be odd. So, (a+b) and (a-b) are both even. 80 can be factorized as a product of even numbers as:

80 = 10*8 = 20*4 = 40*2 So, (a,b) = (9,1), (12,8) or (21,19)

Now we’ll examine each case.

CASE I: (a,b) = (9,1) It means DeVilliers had 1 run at the beginning of the match and reached 81 after it. This case is not possible as he was ranked first prior to the game. CASE II: (a,b) = (12,8) It means DeVilliers had 64 runs at the beginning of the match and reached 144 after it. So the other three players must have had 49, 36 and 25 runs, Gayle being at 25. If we add a perfect square more than hundred to 25, we observe that 25+144=169, which is also a perfect square. But this case is also negated as DeVilliers still led the chart. CASE III: (a,b) = (21,19) It means DeVilliers had 361 runs at the beginning of the match and reached 441 after it. So the other three players must have had 324, 289 and 256 runs, Gayle being at 256. If we add a perfect square more than hundred to 256, we observe that 256+144=400, which is also a perfect square. This case satisfies the given conditions, that DeVilliers still leads the chart and Gayle jumps from the 4th spot to the 2nd spot. Since Maxwell/Miller were at 289 and 324, neither of them scored more than 50 and their new aggregates are still perfect squares, their new aggregates must be 324 and 361. So their scores must have been 35 and 37. So, the total runs scored by these four players in the match = 80+37+35+144 = 296

 Player Before the match Score in the match After the match DeVilliers 361 80 441 Maxwell/Miller 324 37 361 Miller/Maxwell 289 35 324 Gayle 256 144 400

Total = 80 + 37 + 35 + 144 = 296 [i]
[i]