Trick of Mind

Collatz Series

2009-11-10T21:39:00.000-08:00

A strange series of numbers exists where the next number in the sequence depends on the previous one being either even or odd. If it was even, the next is one half of the previous, if odd the next one is 3 times the previous, plus one. The last number of the series is always one. The number of steps to get to one is sometimes called the halting point. (Even the next is N/2; if Odd the next is 3*N+1)

Example, Intital number is 6 then the steps are... 3,10,5,16,8,4,2,1
This has 8 steps and the high number is 16.

(If you plot the initial number vs. the high, you get a neat picture)

There once was a reward of $500.00 for solving various aspects of this series. See

http://en.wikipedia.org/wiki/Collatz_conjecture.

Your question is this. There are six different initial numbers between 3 and 100
that end with these nine steps... 40,20,10,5,16,8,4,2,1 ... what are the six initial numbers?

As a hint, each of these six initial number has 17 steps to reach 1.
So as you can see the first 8 steps vary, but the step nine is 40.

Please give as many of the six you can solve.

Trio is Back

2009-11-06T11:30:00.000-08:00

Tom, Dick and Harry went fishing, having agreed to share the catch equally. After a successful day, they put the catch on the boat deck to be divided up later, had a big party, and went to sleep. In the middle of the night the Tom wakes up, feeling ill, and decides to go home with his share. But he sees that the catch can't be divided evenly into three because there is one fish too many. So he throws the extra fish into the lake, takes his share, and goes home. A little later the Dick wakes up, also feeling sick, and decides to go home with his share. Not realizing that Tom has already left, he also sees that the catch can't be divided equally because of an extra fish. He throws the extra fish into the water and departs. Finally Harry wakes up, ill as well, and unaware that his friends have gone, counts the fish, finds one extra, throws it away and takes his share.

The question is, What is the smallest number of fish that the fishermen could have caught during the day?

Space Elevators

2009-11-05T11:26:00.000-08:00

Armstrong says space elevators must be built. He’s planning to build a carbon nanotube ribbon that is three feet wide, anchored somewhere in Ecuador, stretching a mind-numbing 62,000 miles upwards into space. The centripetal force of Earth’s rotation will keep the ribbon taut. The problem is – if I sneak on to the platform in Ecuador and snip off the ribbon at its base, what will happen?

Circular Coverage

2009-11-04T12:18:00.000-08:00

Consider a disk with a radius of one unit. We need to cover the disk’s surface with some smaller disks so that none of the larger disk can be seen. These smaller disks have a diameter of one unit. How many (the least number) will it take to completely cover the surface?

On Logic

2009-11-04T11:25:00.000-08:00

1) All of the humankind, other than my footmen, have a certain amount of common-sense;
2) No one, who lives on barley-sugar, can be anything but a mere baby;
3) None but a hopscotch player knows what real happiness is;
4) No mere baby has a grain of common-sense;
5) No engine-driver ever plays hopscotch;
6) No footman of mine is ignorant of what true happiness is.

What logical conclusions you can make from here ?

Six and Seven

2009-11-03T11:22:00.000-08:00

How do you proof, for the fact that if any set of integers is repeated six times to form another integer it must be divisible by seven? (eg. 111111)

Sum Problem

2009-11-02T11:21:00.000-08:00

In a group of 10 people, each one is asked to write sum of the ages of the other 9. The sums form the 9 element-set (82, 83, 84, 85, 87, 89, 90, 91, 92). Find the ages of the youngest and the oldest in the group and the ages of the two persons with the same age.

Mirror oh Mirror

2009-11-01T11:19:00.000-08:00

How can you differentiate between a normal mirror and a 1-way mirror (used by spy and police interrogation work by CIA etc, where you can see through from the back) without turning the mirror over?

Water!

2009-10-31T19:37:00.000-07:00

Lies and Statistics

2009-10-29T07:04:00.000-07:00

An independent survey conducted in a large school indicated the lying habits of the students were as follows
category / percentage of students / percentage of lies told of all statements made): (A) Truthful students / 10% / 0%;
(B) Students who seldom lied / 20% / 20%;
(C) Students who evenly lied / 40% / 50%;
(D) Students who frequently lied / 20% / 80%;
(E) Students who were liars / 10% / 100%.

The same group of students were given a multiple choice questionnaire and were asked to indicate the category (A, B, C, D, or E) which they belonged to. Obviously students would indicate their category, depending upon their nature – eg: All truthful students would put themselves in category A, but Liars would never indicate category E.

The problem is, what would be the percentages of students in various categories as self indicated by the students? Further if a second self-assessment survey is conducted asking the students whether they had truthfully answered the first self assessment survey having only two categories (T) Told the truth, and (F) Lied, what would the results tell in percentage terms?

Eight Women

2009-10-28T06:56:00.000-07:00

Eight women, including Ms Irwin, went into a department store to buy one item each. Each went directly to the floor where the item she needed was sold, made the purchase, and left immediately. Two items were sold on each of the store’s four floors (first floor through fourth floor). Can you find each woman's full name (one was named Gracie), the item she bought, the floor it was sold on, and, in the cases of the women who left the first floor and how each arrived at and left the floor of their purchase?

(1) The three women who used the escalator at least once were Eilsa, Ms Kring, and the woman who bought the oil painting;
(2) The dress and the radio were sold on the same floor;
(3) The three women who used the elevator to get both up and down were Dielia (who bought the lamp), Ms Loing, and the woman who bought the radio;
(4) Ms Nielson got her item on the fourth floor;
(5) Claria did not buy the ring. She and Beth went to the same floor;
(6) The three women who did not use the elevator at all were Eilsa, Ms Miller, and the woman who bought the coat;
(7) Helen used both the elevator and escalator;
(8) The woman who bought the linens went down by escalator;
(9) Ms Jonas and Alice took the elevator to their different floors. Flo and the woman who bought the books took the escalator up together, since they were going to the same floor. All four rode down on the elevator at the same time and met Ms Perkins as she entered it on the ground floor;
(10) The only woman to use the stairs was Ms Owens, who did not make her purchase on the third floor.

News

2009-10-28T04:32:00.000-07:00

First the good news: There is no bad news !
Bad News: There is no good news !

How many good news/bad news are there ?

700 MB CD

2009-10-27T22:34:00.000-07:00

I have a file to put on a 700 MB CD.

Windows explorer says the file size is 717,112 KB.
So I expect it's not going to fit.
What's the probability that it WILL fit?

If so, why?

(Boy are things going downhill. I think everyone burned out their brains on the 720!)

Four-stroke motorbike

2009-10-27T06:54:00.000-07:00

In my four-stroke motorbike user’s manual something like this is written in the engine tuning section. "turn the air_fuel mixer control screw fully clockwise to make the mixture rich (with fuel). Now turn the screw anti clockwise two and a half times to make the mixture leaner. The engine RPM will increase. Adjust the idling speed just enough to keep the engine running . . .”

Now the question is, why does the engine runs faster when you supply leaner mixture meaning fuel)?

The Mall Problem

2009-10-26T18:39:00.000-07:00

8 small store occupy the new Great Mall in the shape of a 3x3 grid (with the centre square being an open area) numbered 1 to 8 clockwise from the top left square. They are a bookstore, flower shop, frozen yogurt shop, futon shop, painting and wallpaper store, pizza, photo studio, and shoe store. The 8 proprietors include five women – Eimma, Francis, Ruith, Vecke and Ziina – and three men – Alberto, Georgio and Lukio. The proprietors’ last names are Cole, Gallo, Hanley, Jackson, Klein, Martinez, Riley and Silver. Can you match each shop’s number with the full name of its owner and the kind of shop he or she runs?

(1) On the northern wall are (in no particular order) Eimma’s store, Lukio’s store, and Hanley’s store;
(2) Riley’s shop is situated directly opposite the florist’s, and isn’t adjacent to Cole’s place;
(3) Klein’s shop and the pizzeria aren’t both on the eastern wall;
(4) Francis and Gallo are the only two women who don’t have corner stores;
(5) Neither Alberto’s store nor Jackson’s is on the southern wall;
(6) Georgio’s shop is adjacent to the Bookstore;
(7) On the western wall are (in no particular order) Vecke’s shop, Martinez’s shop, and Ziina’s futon shop;
(8) The pizzeria and the frozen yogurt shop occupy opposite corners of the mall;
(9) The florist is adjacent to the photo developer;
(10) The paint and wallpaper store is adjacent to both Ruith’s shop and the shoe store;
(11) Lukio’s shop is adjacent to Silver’s.

"That's all the information I have Holmes" said Watson.

X Factor

2009-10-25T22:45:00.001-07:00

What is the product of the following series?

(x-a)x(x-b)x(x-c)x(x-d)....(x-z)

-Knightmare

720!

2009-10-24T12:06:00.000-07:00

The value of 720! is a large number.
Just how many digits does it have?
How many of the digits are 0?
How many of the other digits from 1 to 9?

I guess this is to hard.
Here's a link that will make it easy.

http://www.numberempire.com/factorialcalculator.php

a! * b! = c!

2009-10-23T22:18:00.000-07:00

Find three numbers where a is less than b and b is less than c
where a! * b! = c!

Super Women

2009-10-22T02:34:00.000-07:00

What does a woman do everyday, that if a man does once, he dies?

-- Greets from Germany

1997 was a PRIME year

2009-10-20T12:19:00.000-07:00

1997 was a PRIME year....

1997 is the 302nd prime number, and:
- 1997 (prime)
- 1 997 (a unit and a prime)
- 19 97 (two more primes)
199 7 (two more primes)
1997 - (still prime)

So, when is the NEXT year with this property?

Ordered Pairs

2009-10-20T02:19:00.000-07:00

Find all ordered pairs (A,B) such that:

A! = 1680 * (B!)

where "!" denotes the factorial function. For those unfamiliar, the factorial function is defined as the product of all integers less than or equal to the argument and greater than zero. For example, 6! = 6 * 5 * 4 * 3 * 2 * 1. In addition, the argument must be a non-negative integer (i.e. (6.5)! does not exist).

-- Brian Furtado

Urgent Letter

2009-10-19T17:07:00.001-07:00

A military car carrying an important letter must cross a desert. There is no petrol station in the desert, and the car's fuel tank is just enough to take it half way across. There are other cars with the same fuel capacity that can transfer their petrol to one another. There are no canisters or rope to tow the cars.

How can the letter be delivered?

Duelling dilemma

2009-10-19T07:26:00.000-07:00

There are there duellists. One is very good, one is average and the third is rubbish. What is the best strategy for each of them to maximise their chances of surviving in a three way duel. Assume that they are standing at the corners of an equilateral triangle with sides of length 30 feet. The exact detail isn't important, I'm just trying to cause the focus to be on the real logic problem presented.

This is a classic. I haven't disguised it. I haven't even Googled it.

1-9 again

2009-10-18T16:58:00.001-07:00

Find the smallest prime number that contains each digit from 1 to 9
at least once.

1 to 9

2009-10-18T15:36:00.000-07:00

The sum of the nine single digits 1,2,3,4,5,6,7,8,9 equals 45
and the product of the nine digits is 362,880.

Find a different string of nine single digits that have the same sum (45) and product (362,880) as the "123456789" string.

(To be different, some digits will be missing and some will occur more than once)

For example, a string of nine digits could be "112233445" but it would not sum to 45, and the product would not be 362,880, so it meets neither condition.

To easily identify a string, put the digits in increasing order.