數學 problem solving

(a) From O1 and O2, it can be determined that the worst grade is a boy other than Alex, this must be Marvin.
From O1 and O2, Alex is either grade A or B for there are two girls and the worst boy after him.
The potential grade score distributions (in A to E order) are case:
[1] {Ellen, Alex, Candy, Venus, Marvin} or [2] { Alex, Ellen/Venus, Candy, Ellen/Venus, Marvin}

(i) If Alex did not score an A, (case [1])
the grade scored by Alex was B for there were two girls and one boy after him;
the grade scored by Ellen was A for she was not the two girls after Alex and thus before him;
the grade scored by Venus was D for she was one of the two girls after Alex and not Candy.

(ii) If a boy scored an A, (case [2])
this boy was Alex.
The possible grades scored by Ellen and Venus were B or D for the girl other than Candy and worse than Alex was not defined.

(iii) If all the girls did better than the boys, the first observation must be incorrect for Alex did better than girls contradicts with the condition that no boy can be better than girl.

(b)
(i) It will take 49 breaking times (an odd number) until no more breaking is possible. Therefore the first player will also be the last one to break the chocolate, Catherine is the winner.

(ii) The strategy is to maintain each dimension of the small square pieces larger than or equal to 2, otherwise if any dimension reached one piece wide, the next turn player can easily obtain a single piece of the small square pieces by breaking at either end. When the chocolate piece reaches a dimension of 2x2, it will enforce the next player to create a 1x2 chocolate and enable the opposite player to win.

其實根據(O1)-(O4)可以睇到兩樣野
Alex ?
Marvin E (因為女仔冇the worst score, 而男仔中Alex又好過2個或以上女仔, 所以得番Marvin係worst score)
Candy C
Ellen ?
Venus ? (not A)

(a) (i)
咁即係話Alex 會score B(因為唔係A, E同C又已經有人, 而D就冇可能因為佢要好過兩個或以上既女仔)
而Ellen就會係A Venus就會係D
所以answer係
Alex B
Ellen A
Venus D

(a)(ii)好明顯A一定係Alex, 因為Marvin係最差一定唔係A.
咁而家Ellen同Venus既score就有機會B或者D, 因為得番呢兩個grade

(a)(iii) (O1)會錯, 因為全部女仔好過男仔呢個observation會有矛盾

(b)(i)
Catherine 先既話即係代表左佢會拎第1,3,5,7,9.......43,45,47,49塊朱古力
而Christine就會拎第2,4,6,8,10......44,46,48,50塊朱古力
即係Christine會拎最後個塊
Christine win

(b)(ii)
都係Christine win
個strategy就係俾對方break先