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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.

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..

Spoiler

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?

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.

- Maclaurin olympiad & BMO
- Senior Maths Challenge
- Bmo/smc ukmt
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- SMC Prep
- Maths Olympiads Remaining
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- Smc & bmo
- BMO Prep
- Is it too late for serious progress at BMO1/BMO2
- Correlation between UKMT Maths challenge and being able to do Maths at Uni
- SMC preparation
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- Senior maths challenge
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- Mathematics
- SMC-BMO
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