The Enigma Plaza

[Hipmonlee]

Ok I am going to contribute to this thread rather than just answering everything.

Way back in the days of advance, I made up a pokemon based puzzle. The idea was, that there is an arceus attacking you, but you forgot what colour is what type. But this arceus just keeps using judgement repeatedly. What you have to do is send out 3 pokemon, and by only looking at the effectiveness line determine what type the arceus is.

For example! Back when I first asked this question, one answer was Claydol, Aerodactyl, Poliwrath (with water absorb). You could work out the type of any attack just by looking at the super effectiveness response.

However, we already did that, so this time, you have to find an answer using only DP and BW pokemon. Also can you find solutions for BW only or DP only.

I know of a solution for BW only (as does everyone who has been on irc for the last hour or so). There are probably others.

Have a nice day.

 

[Fatecrashers]

A solution for BW (I think): Emboar, Archeops, and Sawsbuck.

EDIT: well shit, this is harder than it looks

 

[Hipmonlee]

Bug and Fire.

[edit] - I just realised I screwed up my solution too..

[edit 2] - ok I found a real BW one.

Have a nice day.

 

[Fatecrashers]

Found one!

Chandelure - Ghost / Fire (Flash Fire)
Hydreigon - Dark / Dragon (Levitate)
Terrakion - Rock / Fighting (Justified)

 

[lati0s]

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I have a large supply of counters, colours red, blue and white. I place one counter on each square of an 8x8 regular chess board. A particular pattern of coloured counters is called an arrangement. Determine whether there are more arrangements with an odd number of red counters or more arrangements with an even number of red counters. (0 is neither even nor odd)

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odd by 2^64-1

Problem two

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I have 6 separate lengths of string. One end is picked at random and is tied to one of the 11 other ends. This process is then repeated until no end of string is left untied. I then count the number of circles of string at the end. What is the average number of circles?

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1/11+1/9+1/7+1/5+1/3+1=1.88

Problem three

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The nonnegative real coefficents a1... a n-1 of the polynomial f(x) = x^n + an-1x^n-1 + ... + a1x+1 are such that all n roots are real. Prove f(3) >= 4^n (for all n>=1)

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are the coefficients supposed to be integers, otherwise it is false.

Problem four

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Two players are playing a number subtraction game using two four-digit numbers. The first player calls out a digit between 0 and 9 (these may be repeated), and the second player places it in any free space.

i.e. abcd-efgh, where each is a digit from 0 to 9, digits can be repeated (e.g. a,d and f can all be 3), and the first player calls out the number in order, but the second player allocates where each number is placed in the subtraction.

The first player tries to make the difference as large as possible, while the second tries to make the difference as small as possible. Assuming both players play to win, find the maximum possible difference.

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if the caller says 0 until a first digit is filled then says 9 continually he can guarantee a difference of at least 9000-0999=8001, this is maximal because the placer can never ensure a number less than 9000 and one greater than 0999

 

[Miscellaneous]

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odd by 2^64-1

correct

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1/11+1/9+1/7+1/5+1/3+1=1.88

Correct

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are the coefficients supposed to be integers, otherwise it is false.

f(x) is monic and has n roots. the coefficients are all real and nonnegative. This is possible. There may be some fault with the way I'm writing the question out (it's much easier with a pen).

My solution:

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as f(x) is monic;
f(x) = (x+α1)(x+α2)...(x+αn) ,,,,,eqn (3)
where -α1...,-αn are the roots of f(x). If all the roots are real then all the α's must also be real.

the coefficients α1...αn-1 are all real and nonnegative. Therefore f(x)>0 for all x>=0. So the roots are all negative, and thereby all the α's are positive.

We also know (because the constant term in the expression for f(x) is 1) that α1α2...αn = 1. We might also speculate at this point that the equality of the given expression when all the α's are equal (to 1).

Substituting x=3 into eqn (3) gives:

f(3) = (3+α1)(3+α2)...(3+αn). ,,,,,,eqn (4)

Applying the arithmetic mean-geometric mean inequality to the collections {1, 1, 1, αi} (motivated by the fact that we are expecting equality when all the α's are equal to 1) gives

3+αi = 1 + 1 + 1 + αi >= 4 sqrt(αi) i = 1...n


so multiplying these n equations together gives


(3+α1)....(3+αn) >= 4^n sqrt(α1α2...αn) = 4^n.


Combining with eqn(4) gives the required result.

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if the caller says 0 until a first digit is filled then says 9 continually he can guarantee a difference of at least 9000-0999=8001, this is maximal because the placer can never ensure a number less than 9000 and one greater than 0999

if the caller calls 0 then:
0xxx-xxxx

There is a way to ensure a number greater than 0999 even with optimal play from the second player.

answer to the subtraction game. It's really tricky, I was told it by a friend in one of our fast-track lessons.

4000 is max. consider 4> and 5<. player 2 will put larger numbers than 5 under and smaller numbers than 4 on top.

method 1: player one calls out a 4 and we get
4xxx-xxxx
follow with 0's to get 4000-0000 = 4000

method 2: player one calls out a 5 and we get
xxxx-5xxx.
follow with 9's to get 9999-5999 = 4000

New question:

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At an annual grand defile, all students of a ballet school appear on stage together in a choreographed curtain call. This year, the 100 students (all distinct heights) are arranged in a square grid. We identify dancer A by finding the tallest student in each column, and selecting the shortest of these 10. We identify dancer B by finding the shortest student in each row, and selecting the tallest of these 10. If A and B are distinct dancers, which is the taller and why?

New question:

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Every day for the next eleven days I shall eat exactly one sandwich for lunch, either a ham sandwich or a cheese sandwich. However, during that period I shall never eat a ham sandwich on two consecutive days. In how many ways can I plan my sandwiches for the next eleven days?

Logic:

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At a house party, 2 guys are having an argument.
Guy 1 says to guy 2: "you eat more food than I do"
guy to guy 1: "you bastard, that's not true"
Girl 1: "you're both wrong"
Girl 2 (to girl 1): "You are correct"
How many true statements are there?

Edit: Answers are below. Thread seems to have died, so PM me if you want any methods

 

[lati0s]

f(x) is monic and has n roots. the coefficients are all real and nonnegative. This is possible. There may be some fault with the way I'm writing the question out (it's much easier with a pen).

My solution:

Hide

as f(x) is monic;
f(x) = (x+α1)(x+α2)...(x+αn) ,,,,,eqn (3)
where -α1...,-αn are the roots of f(x). If all the roots are real then all the α's must also be real.

the coefficients α1...αn-1 are all real and nonnegative. Therefore f(x)>0 for all x>=0. So the roots are all negative, and thereby all the α's are positive.

We also know (because the constant term in the expression for f(x) is 1) that α1α2...αn = 1. We might also speculate at this point that the equality of the given expression when all the α's are equal (to 1).

Substituting x=3 into eqn (3) gives:

f(3) = (3+α1)(3+α2)...(3+αn). ,,,,,,eqn (4)

Applying the arithmetic mean-geometric mean inequality to the collections {1, 1, 1, αi} (motivated by the fact that we are expecting equality when all the α's are equal to 1) gives

3+αi = 1 + 1 + 1 + αi >= 4 sqrt(αi) i = 1...n


so multiplying these n equations together gives


(3+α1)....(3+αn) >= 4^n sqrt(α1α2...αn) = 4^n.


Combining with eqn(4) gives the required result.

woops, I missed the part where the constant term was 1 I read it as being one of the unknown coefficients.

New question:

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At an annual grand defile, all students of a ballet school appear on stage together in a choreographed curtain call. This year, the 100 students (all distinct heights) are arranged in a square grid. We identify dancer A by finding the tallest student in each column, and selecting the shortest of these 10. We identify dancer B by finding the shortest student in each row, and selecting the tallest of these 10. If A and B are distinct dancers, which is the taller and why?

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If A and B are in different columns and rows, let C be the student in the same column as A and the same row as B, then A is taller than the C as it is the tallest in its column, and B is shorter than C as it is the shortest in its row, If A and B share a row or column then A must be taller than B because A is the tallest in its column and B is the shortest in its row.

New question:

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Every day for the next eleven days I shall eat exactly one sandwich for lunch, either a ham sandwich or a cheese sandwich. However, during that period I shall never eat a ham sandwich on two consecutive days. In how many ways can I plan my sandwiches for the next eleven days?

Hide

233

Logic:

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At a house party, 2 guys are having an argument.
Guy 1 says to guy 2: "you eat more food than I do"
guy to guy 1: "you bastard, that's not true"
Girl 1: "you're both wrong"
Girl 2 (to girl 1): "You are correct"
How many true statements are there?

Hide

the girls are both definitely lying and one of the guys is wrong so exactly 1 statement is true