Roopesh - The ice cream eater
Roopesh went to his friends house. There he ate a lot of Ice-cream. First he started with Vanilla then Chocolate then Vanilla then Butterscotch then Vanilla. He is a mathematician and made a monic polynomial of degree 5 which gave him values of the first letter of the Ice Cream.
This is the way his polynomial proceeded :-
p
(
1
)
=
2
2
p
(
2
)
=
3
p
(
3
)
=
2
2
p
(
4
)
=
2
p
(
5
)
=
2
2
If the value of p ( 6 ) = a b c . The first letter of the ice cream he eats would be a + b + c . Which ice cream will be eat next?
Assumption: a b c represents a 3 digit number.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
2 solutions
A no-brainer method would be to use the Method of Finite Differences . Since it is given that p ( x ) is monic degree 5 polynomial, you have,
D 5 ( n ) = 5 ! = 1 2 0 ∀ n ∈ Dom ( p )
Here's the relevant part of the constructed difference table:
n 1 2 3 4 5 6 p ( n ) 2 2 3 2 2 2 2 2 4 3 7 D 1 ( n ) − 1 9 1 9 − 2 0 2 0 4 1 5 D 2 ( n ) 3 8 − 3 9 4 0 3 9 5 D 3 ( n ) − 7 7 7 9 3 5 5 D 4 ( n ) 1 5 6 2 7 6 D 5 ( n ) 1 2 0
Log in to reply
cool, never knew about this! this would have been useful to know...
Best solution!
When you write that the polynomial roots are 1, 3, and 5, wouldn't that be (x-1)(x-3)(x-5)(x-a)(x-b)? I noticed you did the math correctly regardless, just some possible confusion there...
Very nice. Now I see why this was a good question.
I did exactly the same.
Let G(x) = P(x)-22 = (x-1)(x-2)(x-3)(x-4)(x-5) + h(x) Since G(x) is monic this implies that deg(h(x))<=4 Since h(1)=h(3)=h(5)=0 Let h(x)=A(x-1)(x-3)(x-5)(x+b) Using the fact that h(2)= 19 and h(4) = -20 Values for A and B can be gotten to be A = 6.5 and b=-116/39 Then h(6) = 6.5(5)(3)(1)(6-(116/39))= 295 Then G(6)= 5! + h(6) = 415 Finally we see P(6) = G(6) + 22 = 437 4+3+7=14 Which tells us that our Ice cream must begin with letter "N"
Let Q ( x ) = P ( x ) − 2 2 1 , 3 and 5 are the roots of Q ( x ) , It follows that Q ( x ) = P ( x ) − 2 2 = ( x − 1 ) ( x − 3 ) ( x − 5 ) ( x − a ) ( x − b ) Note that Q ( 2 ) = P ( 2 ) − 2 2 = − 1 9 = 3 ( 2 − a ) ( 2 − b ) Q ( 4 ) = P ( 4 ) − 2 2 = − 2 0 = − 3 ( 4 − a ) ( 4 − b ) Last two equations give us: a + b = − 2 1 a b = − 3 3 4 Now we have P ( x ) = 2 2 + ( x − 1 ) ( x − 3 ) ( x − 5 ) ( x − a ) ( x − b ) = 2 2 + ( x − 1 ) ( x − 3 ) ( x − 5 ) ( x 2 − ( a + b ) x + a b ) = 2 2 + ( x − 1 ) ( x − 3 ) ( x − 5 ) ( x 2 + 2 1 x − 3 3 4 ) It gives P ( 6 ) = 4 3 7