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BMO question...
17
Alice and Barbara play a game with a pack of 2n cards, on each of
which is written a positive integer. The pack is shuffled and the cards
laid out in a row, with the numbers facing upwards. Alice starts, and
the girls take turns to remove one card from either end of the row,
until Barbara picks up thefinal card. Each girl's score is the sum of
the numbers on her chosen cards at the end of the game.
Prove that Alice can always obtain a score at least as great as
Barbara's.
which is written a positive integer. The pack is shuffled and the cards
laid out in a row, with the numbers facing upwards. Alice starts, and
the girls take turns to remove one card from either end of the row,
until Barbara picks up thefinal card. Each girl's score is the sum of
the numbers on her chosen cards at the end of the game.
Prove that Alice can always obtain a score at least as great as
Barbara's.
Reply 1
why is it important that it is 2n...why is it important that Alice starts first? What is the significance of the terminology 'at least as good'? coincidentally, I was showing this to some of my school colleagues/teachers a few days ago..
Reply 2
Spoiler
Reply 3
Chewwy
OP
17
cool.
Reply 4
Can't you just say:
Observe that the game can be split into n rounds, where one round involves Alice and Barbara respectively picking 1 card from either end of the row of cards.
Surely either the cards have the same value, of one is greater than the other. Since Alice goes first, every round she has the chance to prevent Barbara picking a card of greater value than hers.
--------------
Have I missed something?
Alice doesn't have to pick the same end of the row each round does she?
Observe that the game can be split into n rounds, where one round involves Alice and Barbara respectively picking 1 card from either end of the row of cards.
Surely either the cards have the same value, of one is greater than the other. Since Alice goes first, every round she has the chance to prevent Barbara picking a card of greater value than hers.
--------------
Have I missed something?
Alice doesn't have to pick the same end of the row each round does she?
Reply 5
Chewwy
OP
17
C4>O7
Can't you just say:
Observe that the game can be split into n rounds, where one round involves Alice and Barbara respectively picking 1 card from either end of the row of cards.
Surely either the cards have the same value, of one is greater than the other. Since Alice goes first, every round she has the chance to prevent Barbara picking a card of greater value than hers.
--------------
Have I missed something?
Alice doesn't have to pick the same end of the row each round does she?
Observe that the game can be split into n rounds, where one round involves Alice and Barbara respectively picking 1 card from either end of the row of cards.
Surely either the cards have the same value, of one is greater than the other. Since Alice goes first, every round she has the chance to prevent Barbara picking a card of greater value than hers.
--------------
Have I missed something?
Alice doesn't have to pick the same end of the row each round does she?
lol, this is what i thought to begin with. the thing is, consider if the numbers were like this:
1,1,1000,2
using your tactics, you'd lose. badly.
Reply 6
Oh I see now
Barbara now has a new card to choose once Alice takes hers...
Barbara now has a new card to choose once Alice takes hers...
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