Puzzles with numbers and things

[davis_greatest]

snookersfun extends her lead

hopefully 970, after a bit more thought.

970 is the correct answer. Well done!

Let S = the Special Number

and davis_greatest’s answer be Davis = S + 1/S
and Gordon’s answer be Gordon = S^2 + 1/S^2
and Oliver’s answer be Oliver = S^3 + 1/S^3

Then Davis^2 = (S + 1/S)^2 = S^2 + 1/S^2 + 2 = Gordon + 2
So Gordon = Davis^2 – 2

Gordon has the same number of digits as Davis.
Davis has more than one digit, so Davis >=10
If Davis = 10, the Gordon = 10^2 – 2 = 100 – 2 =98 which has the same number of digits as Davis.
If Davis>=11, then Gordon = Davis^2 – 2 has more digits than Davis.

So Davis = 10.

Once we know this, there is more than one way of getting Oliver’s answer. One way is to note that
Davis^3 = (S + 1/S)^3 = S^3 + 1/S^3 + 3S + 3/S = Oliver + 3 Davis

So Oliver = Davis^3 – 3 Davis = 10^3 – 3x10 = 1000 – 30 = 970



SO HERE IS THE SCOREBOARD AFTER ROUND 31

snookersfun……………………….…..15
Vidas……………………………………….8½
robert602…………………………………5
abextra……………………………..…...4½
davis_greatest…………………..……3

(some rounds may be worth more than one point)
(especially ones won by davis_greatest)

[elvaago]

Why are you subtracting 2 and 30 from respectively 100 and 1000? I don't see anywhere in your problem that you have to deduct anything from anything.

[rambon]

You've lost me for the first time I think.

If you add the reciprocal of 10 (1/10) to 10, you get 10.1 don't you? Why is that a whole number. I've just spent eight hours on a plane, so I admit that jetlag might have played a role in this.....

[elvaago]

I think he means that S + 1/S = 10, where S is not 10, but more something like, 9.89.

[davis_greatest]

I think he means that S + 1/S = 10, where S is not 10, but more something like, 9.89.

Precisely. You don't have to know the Special Number S to solve the problem. But in fact S + 1/S = 10, so S could be either about 9.899 or about 0.101.

[davis_greatest]

Why are you subtracting 2 and 30 from respectively 100 and 1000? I don't see anywhere in your problem that you have to deduct anything from anything.

What we know is that Charlie tells davis_greatest a Special Number S, which I add to its reciprocal to get
S + 1/S. Call this answer "Davis".

Charlie tells Gordon the square of the special number S^2, which Gordon adds to its reciprocal to get
S^2 + 1/S^2. Call this answer "Gordon".

Charlie tells Oliver the cube of the special number S^3, which Oliver adds to its reciprocal to get
S^3 + 1/S^3. Call this answer "Oliver".

With some algebra (shown above) - regardless of the value of the special number S - you get:

Gordon = Davis^2 – 2

and

Oliver = Davis^3 - 3 x Davis.

That's why you don't need a calculator - because at the end all you need to calculate is
10 x 10 x 10 - 3 x 10 = 1000 - 30 = 970.

[snookersfun]

I feel smoking heads... and I don't feel so silly anymore

[davis_greatest]

I feel smoking heads... and I don't feel so silly anymore

Hehe. Did you solve it the same way?

[snookersfun]

Haha, of course not. Why would I have had a range otherwise?
got to the 10 in about the same way (only possibility to get same # of digits), after that figures that the number has to be around 9.9 (bit smaller in fact), therefore 2nd answer 98 and third 970 (as really tiny numbers are added to those, they couldn't have been the next highest numbers)

[davis_greatest]

Are you satisfied with the question and answer now, elvaago and rambon?

[elvaago]

I'm not especially bright today, I'll take your word for it. I'll finish with the following gem, though:

Assuming that there is a number i for which goes i^2 = -1 you can deduce the following:

-1 = i^2 = i * i = square root of -1 * square root of -1 = square root of (-1 * -1) = square root of (-1^2) = square root of 1 = 1

Therefore, -1 = 1

[davis_greatest]

Why mess around with i? With that logic, you could just as well say:

-1 = square root of (-1) squared = square root of 1 = 1

[elvaago]

Because everyone will see in your example that square root of -1 is wrong, but if you confuse them with i most people will be confused. :-)

[davis_greatest]

Because everyone will see in your example that square root of -1 is wrong, but if you confuse them with i most people will be confused. :-)

I am not taking square root of -1... what I meant was

-1 = square root of [ (-1) squared ] = square root of [1] = 1

[davis_greatest]

Round 32 - A load of balls

ROUND 32

I am playing snooker with Charlie. The six colours are left, Charlie comes to the table, and I tell him that I’ll give him a penny for every point in his next break. Sure enough, he clears the six colours and I generously give him his money. He adds this to the £100 he already has in his top pocket, and goes to buy some snooker balls from Barry The Baboon’s Ball Shop. (Each ball costs a whole number of pence and they can be bought individually, each costing the same. He buys one white ball and a whole load of balls of other colours.)

When Charlie gets home, he decides to play with his balls. He places the white ball in the middle of the snooker table, and then places a first ring of balls around it, as closely as possible, with each ball in the ring touching the white ball and the adjacent balls in the ring. Then he puts a second ring around the first ring, again as closely as possible, so each ball in the 2nd ring touches the first ring and the adjacent balls in the ring. And so he continues, adding rings, using up all his balls exactly.

How much did each ball cost?