My Neurons Love Me For Exercising Them
Yesterday’s Problem:
If you toss a die six times, what are the chances that all six faces will turn up?
Answer:
There is less than a 2% chance that all six faces will shown up:
6/6 x 5/6 x 4/5 x 3/6 x 2/6 x 1/6 = .015
Today’s Puzzle:
Why is a game of tennis like a party of children?
Last Problem:
A girl throws four balls randomly into any or all of four boxes. She always succeeds in getting the ball into one of the four boxes. But of course, it is possible that some boxes may have more than one ball. What are the chances that after she throws four balls, each box will each contain a single ball?
Answer:
To answer the problem, you will want to simply multiply that chances that each ball will land in an empty box.
Here is what the calculations will look like:
4/4 x 3/4 x 2/4 x 1/4 = 6/64 = 0.09
This means that there is roughly one chance in 10 that each of the four boxes will contain one ball.
Today’s Problem:
If you toss a die six times, what are the chances that all six faces will turn up?
Last Problem:
You see three faces on each of three dice for a total of nine faces. If the sum of the dots on each die is different and you see a total of forty dots altogether, then which faces (showing which numbers) must be visible on each of the three dice?
Answer:
The largest sum you can conceivable see on any dice is 15; that is, the sum of 4,5,6. Therefore, the only possible combinations of three different numbers that total 40 are 15+14+11 and 15+13+12. But, a sum of 13 is impossible to see on three faces of a dice. (If you do not believe me, try it out for yourself). This leaves the only answer as 15, 14 and 11.
This means that one dice will show 6, 5 and 4 dots on the three sides (which equals 15); one dice will show 6, 5 and 3 dots (which equals 14); And, one dice will show 6, 3 and 2 dots (which equals 11).
Today’s Problem:
A girl throws four balls randomly into any or all of four boxes. She always succeeds in getting the ball into one of the four boxes. But of course, it is possible that some boxes may have more than one ball. What are the chances that after she throws four balls, each box will each contain a single ball?
Last Problem:
Read the following sentence:
FINISHED FILES ARE THE RESULT OF YEARS
OF SCIENTIFIC STUDY COMBINED WITH THE
EXPERIENCE OF YEARS.
Now, read it again. this time counting every F you see. How many did you find?
Are you sure you found them all?
Answer:
This is actually a classic game. There are six F’s. The Fs in the word “of” are very easy to miss, so that the count you derive is 3 rather than 6.
Today’s Problem:
You see three faces on each of three dice for a total of nine faces. If the sum of the dots on each die is different and you see a total of forty dots altogether, then which faces (showing which numbers) must be visible on each of the three dice?
Last Problem:
There are three types of people in the city of Las Wages: those who always tell the truth; those who always lie and those who alternate between lying and truth telling.
If you meet an inhabitant on the streets of Las Wages and can ask only two questions, what two questions will enable to you determine what sort of a person he or she is?
Answer:
You should ask the same question twice: “Am I in Las Wages?” and “Am I in Las Wages?
Two yeses will come from a truth teller; two nos will come from a liar. A yes and a no answer (or a no and a yes answer) to the same question will mean that the person alternates between telling the truth and telling a lie.
Today’s Problem:
Read the following sentence:
FINISHED FILES ARE THE RESULT OF YEARS
OF SCIENTIFIC STUDY COMBINED WITH THE
EXPERIENCE OF YEARS.
Now, read it again. this time counting every F you see. How many did you find?
Are you sure you found them all?
Yesterday’s Problem:
You are happily on your way to Truth City where it is well known that all the inhabitants always tell the truth. At one point in your travels you reach a fork in the road. One leg takes you to Truth City; the other leg takes you to Lies City which all the citizens always lie. The sign is very confusing – so you can not tell which fork takes you to Truth City.
A man stands near the sign, so you can ask him for directions. The problem of course is that you do now know whether the man is from Truth City (where everyone always tells the truth) or Lies City (where everyone always lies).
If you have time to ask this man one question and only one question, what can you ask him that will give you a 100% guarantee that you will get the information you need to take the fork which leads to Truth City?
Answer:
Ask the man: Which Way To Your Hometown?
If he is from Truth City he will point to the fork in the road that will take you to Truth City. If he is from Lies City, he will also point to the fork in the road that leads to Truth City (since people from Lies City always lie).
Today’s Problem:
There are three types of people in the city of Las Wages: those who always tell the truth; those who always lie and those who alternate between lying and truth telling.
If you meet an inhabitant on the streets of Las Wages and can ask only two questions, what two questions will enable to you determine what sort of a person he or she is?
Last Problem:
Think of a number – any number. If it is odd, triple it and add 1; If it is even divide it by 2.
Apply this rule to each new number you obtain. What eventually happens? Does this happen with any number you might start with, regardless whether the starting number is even or odd?
(Warning: You could conceivably be working on this problem on day long!)
Answer:
If you begin experimenting with small numbers, it quickly becomes apparent that the above sequences get stuck in a loop of 1-4-2-1-4-2 etc.
If you used numbers larger than 26, you might have found it took much longer to settle into this hamster wheel rut, but it does eventually with any number you start with. If you started with the number 27, the calculations will take you all the way up to 9.232 just when you think this will never settle into the 1-4-2- rut. You will reach the 1-4-2- sequence at step 111. I hope you did not spend all day calculating with larger numbers to see what would happen! Every number up to a trillion has been tested and every number eventually collapses into the same rut. This problem is known in mathematics as the hailstorm problem.
As an analogy – we all have certain seed thoughts which hold us down from actualizing our destiny. It doesn’t matter where we start, we will always wind up connecting with the same seed thought until it is released, removed. detached, ejected and shielded. This is what makes sustainable recovery possible.
Today’s Problem:
You are happily on your way to Truth City where it is well known that all the inhabitants always tell the truth. At one point in your travels you reach a fork in the road. One leg takes you to Truth City; the other leg takes you to Lies City which all the citizens always lie. The sign is very confusing – so you can not tell which fork takes you to Truth City.
A man stands near the sign, so you can ask him for directions. The problem of course is that you do now know whether the man is from Truth City (where everyone always tells the truth) or Lies City (where everyone always lies).
If you have time to ask this man one question and only one question, what can you ask him that will give you a 100% guarantee that you will get the information you need to take the fork which leads to Truth City?