[A boy playing with his balls]

all identical except colour. 8 blue, 4 red

[A boy playing with his balls]

What if the question was, no. of ways you can form a ring with the balls?

[The sums of fourth power numbers]

If cubic equation x³ − Ax² + Bx − C = 0 has roots a, b, c, then, expanding (x − a)(x − b)(x − c), we find

A = a + b + c

B = ab + bc + ca

C = abc

Then B = ab + bc + ca = ½ [(a + b + c)² − (a² + b² + c²)] = 14.

Hence a, b, c are roots of x³ − 6x² + 14x − C = 0, and we have

a³ − 6a² + 14a − C = 0

b³ − 6b² + 14b − C = 0

c³ − 6c² + 14c − C = 0

Adding, we have (a³ + b³ + c³) − 6(a² + b² + c²) + 14(a + b + c) − 3C = 5 − 6×8 + 14×6 − 3C = 0.

Hence C = 41/3, and x³ − 6x² + 14x − 41/3 = 0.

Multiplying the polynomial by x, we have x⁴ − 6x³ + 14x² − 41x/3 = 0. Then

a⁴ − 6a³ + 14a² − 41a/3 = 0

b⁴ − 6b³ + 14b² − 41b/3 = 0

c⁴ − 6c³ + 14c² − 41c/3 = 0

Adding, we have (a⁴ + b⁴ + c⁴) − 6(a³ + b³ + c³) + 14(a² + b² + c²) − 41(a + b + c)/3 = 0.

Hence a⁴ + b⁴ + c⁴ = 6×5 − 14×8 + (41/3)×6 = 0.

That is, the sum of the fourth powers of the numbers is 0.

With this method we find abc=41/3. Now if we expand (a+b+c)^4 as (a+b+c)^2*(a+b+c)^2, then we get a^4+b^4+c^4=24*abc-328, which on substituting abc becomes = 0.

[Airline tickets]

Male surnames are Foley and Hall who is not Tom:

Tom Foley

Bob Hall

Alice is 2 or 3, but Alice=3 leads to Tom = Dallas = 4 which is not permitted.

The other clues lead to two sets of possibilities:

1 Je Lo Elkins Atlanta 1 Je Lo Elkins Atlanta

2 Alice Dawes Chicago 2 Alice Girard Chi Sea

3 Tom Foley StPaul 3 Tom Foley Chi StP Sea

4 Je Lo Girard Dallas 4 Je Lo Dawes Dallas

5 Bob Hall Seattle 5 Bob Hall Chi StP

I don't see contradiction for any of the above remaining cases.

The only way to distinguish Jean from Lois is that Jean does not go to Seattle.

That clue does not seem to help since in any case she seems headed for Atlanta or Dallas

If anyone can eliminate any of the above cases, feel free to carry it out to completion.

If we try to find which place will the 5th person wants to go, then we find that as per Clue 1 - It can't be Seattle or St. Paul. As per Clue3 & 4 - Atlanta and Dallas respectively also can't be, then 5th person should go to Chicago.

[A boy playing with his balls]

Are the balls identical? If not, i think this should work, though not sure.

8C4 / 12C4

The balls are identical in size.

My answer was wrong, anyhow. WIll give a shot soon.

[A boy playing with his balls]

Are the balls identical? If not, i think this should work, though not sure.

8C4 / 12C4

[The sums of fourth power numbers]

Interesting result!

I don't solve cubics. I am retired, and I don't have to do things like that.

Or Bayes formulas, at least until I learn to understand them.

Newton-Raphson, on the other hand is a convenient and general

sledge-hammer tool, and it's fairly simple to program.

I did a manual search that convinced me the numbers

are not integers, and they are not (all) real.

NR tells me that are of the form

a = real

b = x + iy

c = x - iy

With values of

a = 2.67770427706

x = 1.66114786147

y = 1.53116371279

These values gave 6 8 and 5 for sums squares and cubes, within 10^-8.

The sum of their 4th powers? Zero!

I found an alternative and interesting triplet that someone might want to solve.

It may be quite easy because [white spoiler] a, x and y are all integers [/white spoiler].

Hint: to see the spoiler, just select it.

a + b + c = 4

a²+ b²+ c² = 4

a³+ b³+ c³ = 4

a⁴+ b⁴+ c⁴ = ___?

It appeared to me too that a,b,c all cant be real. If we could find the value of a*b*c, then the problem could be done. Can you please provide a link for newton raphson method (some easy explanation )

[The ABC's of numbers]

I couldn't figure out 'I' but most probably F, which is 95, is the largest.

G is 93, and the 2nd largest. H can have multiple values but all less than F.

[One Up Me]

Sorry for such delay. Here goes - 'Going, Going, Gone....'

[Airline tickets]

I am getting

Alice Dawes Seattle - 2nd

Tom Foley St. Paul - 3rd
Bob Hall Chicago - 5th
Lois and Jean can be interchanged - The one who is given Elkins surname goes to Atlanta and is 1st in row, the other goes to Dallas and is 4th in row.

[The sums of fourth power numbers]

Any Hint for me?

[Diamond Subtraction]

I cheated; with less than 6 million different starting combinations, I wrote some code to brute force the answer.

1

8 45

21

Lasted for 12 iterations (possibly 11 or 13, counting isn't my strong suit )

Can you share the code?

[Max product of sums that make 100]

36 times e i.e. e^36, multiplied by e^(100-36*e)

The product would be = e^100/e

[Two Ten Ton Tough Elephants]

11.0527 tons?

Did we write our decimals out of order? Because I got 11.05727... The difference is small though so I will still give you best answer.

90% of 110% of any number n is (.90)(1.10)(n) = 0.99 n . Same goes for 110% of 90%. So each year the elephants lose 1%; we multiply by 0.99 ten times, and we get (if n = orig. wt):

((0.99)^10) * n = 10 tons ;

n = 10 / (0.99)^10

= 10 / 0.904382075

= 11.05727355 tons.

I did the same way as you have done, its just that i thought if i am right, writing till only 3-4 decimal places would be sufficient..

you got .0527 where i got .0572. So i wasn't sure if you used a different strategy or inverted your numbers

Whoops, sorry. I mistyped.

[Two Ten Ton Tough Elephants]

11.0527 tons?

Did we write our decimals out of order? Because I got 11.05727... The difference is small though so I will still give you best answer.

90% of 110% of any number n is (.90)(1.10)(n) = 0.99 n . Same goes for 110% of 90%. So each year the elephants lose 1%; we multiply by 0.99 ten times, and we get (if n = orig. wt):

((0.99)^10) * n = 10 tons ;

n = 10 / (0.99)^10

= 10 / 0.904382075

= 11.05727355 tons.

I did the same way as you have done, its just that i thought if i am right, writing till only 3-4 decimal places would be sufficient..

[Two Ten Ton Tough Elephants]

@brahmaraj - I may be wrong but i think there is mistake in your calculation, our answers are coming same.

[Two Ten Ton Tough Elephants]

11.0527 tons?

[An oldie a professor once told me on Area and Perimeter]

^ On secons thought, post 3 may not be right.

[An oldie a professor once told me on Area and Perimeter]

Further

This is just a guess, since Jen's 1st statement was enough for Ben to determine the numbers, i think, the number of ways 2 integers can be multiplied to get the area, should be only 2, if we apply the constraint that L+W<100.

[An oldie a professor once told me on Area and Perimeter]

Let me do a bonanova -

For starters

L ans W both can't be prime numbers, otherwise Ben could easily find out the values of L & W.

[Calling all positive integers ...]

Agree with Superprismatic. I think u have counted (1, 2*10^250-1) also.

[Oh no I lost my baby!]

Bob - Betty - Larson - Jennifer

Jim - Susan - Jacobs - Ann
Gary - Judy - Miller - Andrew
Chuck - Mary - Smith - Jason
Tom - Carol - Thompson - Richard
George - Trudy - Wilson - Jane and Amy

[A paradox with derivatives]

above poster seems right. It is not right to write f(x)=x+x+x+.....(x times) as x is a variable, so the number of times x comes is also not defined, till we give x a value.

[A Car Travels]

273

[Matching values of Die throws pt. 3]

Ok thanks. I didnt realise expected value may be a fraction.