Number Tricks

Ms. Arroyo asked the class to see if they could find the sum of the first 50 odd numbers. As everyone settled down to their addition, Terry ran to her and said, "The sum is 2,500." Ms. Arroyo thought, "Lucky guess," and gave him the task of finding the sum of the first 75 odd numbers. Within 20 seconds, Terry was back with the correct answer of 5,625.

How does Terry find the sum so quickly?

 

Answer:

The following pattern holds: The sum is equal to n x n, when n is the number of consecutive odd numbers, starting with 1. For example, the sum of the first 3 odd numbers is equal to 3 x 3, or 9; the sum of the first 4 odd numbers is equal to 4 x 4, or 16; the sum of the first 5 odd numbers is equal to 5 x 5, or 25; and so on.
 

The Father of Algebra

Diophantus was a Greek mathematician who lived in the third century. He was one of the first mathematicians to use algebraic symbols.

Most of what is known about Diophantus's life comes from an algebraic riddle from around the early sixth century. The riddle states:

Diophantus's youth lasted one sixth of his life. He grew a beard after one twelfth more. After one seventh more of his life, he married. 5 years later, he and his wife had a son. The son lived exactly one half as long as his father, and Diophantus died four years after his son.

How many years did Diophantus live?

 

Answer:

The riddle, the "facts" of which may or may not be true, results in the following equation:
x/6 + x/12 + x/7 + 5 + x/2 + 4 = x
where x is Diophantus's age at the time of his death.
Therefore, Diophantus lived exactly 84 years.

How Far

After visiting my Great Aunt Annie, I travelled home in her old jalopy. The car was old and battered, it had a leak from the petrol tank, and I was stuck in second gear.

This meant that I could only travel along at a steady 30 miles per hour and managed a paltry 20 miles per gallon of fuel.

At the start of the journey I had placed exactly 10 gallons of fuel into the tank. I knew though, that the fuel tank lost fuel at the rate of half a gallon per hour.

Just as I arrived home, the car stopped because it had run out of fuel and I had only just made it.

 How far was it from my Great Aunt's to my home?


 

Answer:

150 miles.

I was travelling at 30mph at a rate of 20mpg, so I was using 1.5 gallons every hour for the driving itself. I was also losing 0.5 gallon every hour, so in total, I was using 2 gallons every hour. Therefore to use all of the 10 gallons I started with, I travelled for 5 hours.

5 hours at 30mph is 150 miles.

 

January 1

Some people believe that January 1, 2000 is the first day of the 21st century. Other people believe that the honor belongs to January 1, 2001. But everyone should agree that January 1, 2002 is the first "sum-day" of the new century- when you write out that date in standard notation, it becomes 01/01/02, and 1+1=2. More generally, a sum-day is a date in which the day and month add up to the year. With that in mind:

 A) What is the last sum-day of the 21st century?
 B) How many sum-days are there in the 21st century?



 

Answer:

A) The last sum-day of the 21st century is December 31, 2043, because 12+31=43, and both the month and day are as big as possible.

B) This one is much easier than you might think. The correct answer is 365, because every day in a standard (non-leap) year is part of a sum-day for some year. For example, November 26 is a sum-day for the year 2037, because 11+26=37. The only date for which this principle doesn't work is the leap day, February 29. That's because 2+29=31, but 2031 is not a leap year.

Grandmas and Trolls

You are on your way to visit your Grandma, who lives at the end of the valley. It's her birthday, and you want to give her the cakes you've made. Between your house and her house, you have to cross 7 bridges, and as it goes in the land of make believe, there is a troll under every bridge! Each troll, quite rightly, insists that you pay a troll toll. Before you can cross their bridge, you have to give them half of the cakes you are carrying, but as they are kind trolls, they each give you back a single cake. How many cakes do you have to leave home with to make sure that you arrive at Grandma's with exactly 2 cakes?

 

Answer:

2: At each bridge you are required to give half of your cakes, and you receive one back. Which leaves you with 2 cakes after every bridge.

Identify Heavy Balls!

Three out of six look alike balls are heavy. Identify the heavy balls by three times weighing on a beam balance!

Answer:

Number the balls: 1, 2, ..., 6. Then weigh (No. 1, No. 2) vs (No. 4, No. 5), next weigh (No. 2, No. 3) vs (No. 5, No. 6), and then (No. 3, No. 1) vs (No. 6, No. 4).

How much money do Todd and Steve have?

Todd and Steve's money are such that if Todd receive $140 from Steve they will be in the ratio 5 to 1, but if Steve receives $64 from Todd, their ratio will be 1 to 2. How much money do Todd and Steve have in their pockets?

Answer:

Todd: $200

Steve: $208

Object Weights

If we assume that the distance from the center has no effect on the result, which object weights the least?

Answer:

The square weights the least.

Complete a Word!

Fill in a letter to complete a 12-letter word reading either clockwise or counterclockwise. What is this 12-letter word?

________________________

ANSWER: Correlatable (what can be correlated; e.g., correlatable phenomena).
________________________

Find a 14-letter English Word

In the following puzzle, as in the two previous posts (1, 2), starting from any letter, trace out a 14-letter English word by going along the lines without crossing any letter more than once.

________________________


 ANSWER: Dissymmetrical
________________________

Find a 13-letter English Word

In the following puzzle, as in the previous post, starting from any letter, trace out a 13-letter English word by going along the lines without crossing any letter more than once.

_______________________


 ANSWER: Breaststroker
_______________________

Word Ladder: Chase to Catch!

In this word ladder, you should start from Chase and end to Catch by adding 8 words in between!

Note: A Word Ladder is a word game invented by Lewis Carroll. A word ladder puzzle begins with two words, and to solve the puzzle one must find a chain of other words to link the two, where at each step the words differ by altering a single letter [for more info visit http://en.wikipedia.org/wiki/Word_ladder].

________________________


 ANSWER: 
________________________

source: http://www.puzzlechoice.com

Find a 12-letter English Word

In the following puzzle, starting from any letter, trace out a 12-letter English word by going along the lines without crossing any letter more than once.

Find a 12-letter English Word

_______________________


 ANSWER: Tastefulness
_______________________

Little Helpers

Santa is getting ready for Christmas - sorting presents, filling sacks and working out his route to make sure he can deliver all his wonderful presents to the children of the world in time for when they wake up Christmas morning. The trouble is, every year just before Christmas, the reindeer have nothing to do and mostly stand around getting rather bored. They are always asking if they can help but there's nothing they can do until Santa's ready to go.
?This year, Santa is prepared for them and he's set them a little challenge to help keep them busy. He's given them the following problem:
 Five children from five different families living in five different counties of England have asked for a different gift each. From the clues, the reindeer have to work out who's who, where each child lives and what present each has asked for. The first reindeer to solve the problem gets an extra portion of Christmas pudding on Christmas Day!

 1. Neither Cliff, nor Jennifer Feather (who does not live in Yorkshire), lives in Kent.
 2. Young Crawford, who is neither Sarah nor Cliff, asked for the pony (a gift that Santa would have particular difficulty getting down the Crawford family's chimney!).
 3. Young Rowlands is neither the child who lives in Yorkshire nor the child who has asked for a bicycle.
 4. The gift due for delivery to Cornwall, which is not for the child surnamed Rowlands, is a computer.
 5. Liz, who has asked for the painting set, is not from Kent. Her surname is not Jamison.
 6. The child who lives in Essex has asked for a guitar. Alan lives in Cumbria.

Children’s first names: Alan, Cliff, Jennifer, Liz, Sarah
Children’s last names: Crawford, Feather, Jamison, Northey, Rowlands
Counties: Cornwall, Cumbria, Essex, Kent, Yorkshire
Presents: Bicycle, Computer, Guitar, Painting Set, Pony



 

Answer:

 Name              County     Present Alan Crawford     Cumbria    Pony
 Cliff Rowlands    Essex      Guitar
 Jennifer Feather  Cornwall   Computer
 Liz Northey       Yorkshire  Painting Set
 Sarah Jamison     Kent       Bicycle

19

Can you circle exactly three numbers to make a total of 19?

 

Answer:

12 + 6 + 1 = 19.
 Note that the puzzle didn't ask for three digits to be circled!

Toward a Target

You can imagine an arrow in flight, toward a target. For the arrow to reach the target, the arrow must first travel half of the overall distance from the starting point to the target. Next, the arrow must travel half of the remaining distance.

For example, if the starting distance was 10m, the arrow first travels 5m, then 2.5m.

If you extend this concept further, you can imagine the resulting distances getting smaller and smaller. Will the arrow ever reach the target?

 

Answer:

Yes.
Since the arrow does indeed hit the target, it must be true that 1/2 + 1/4 + 1/8 + ... = 1.
This is because the sum of an infinite series can be a finite number.

Brain Workout: Variety Puzzles to Boost Your Memory and Brainpower

Room Rate

Three people check into a hotel. They pay £30 to the manager and go to their room. The manager suddenly remembers that the room rate is £25 and gives £5 to the bellboy to return to the people. On the way to the room the bellboy reasons that £5 would be difficult to share among three people so he pockets £2 and gives £1 to each person. Now each person paid £10 and got back £1. So they paid £9 each, totalling £27. The bellboy has £2, totalling £29. Where is the missing £1?

 

Answer:

We have to be careful what we are adding together.
Originally, they paid £30, they each received back £1, they now have only paid £27. Of this £27, £25 went to the manager for the room and £2 went to the bellboy.

Brain Puzzlers' Bog Snorkelling

Four friends were competing in the internationally renowned Brain Puzzlers' Bog Snorkelling competition. As usual, the judges were a little careless and once again, they managed to loose the results. Luckily, a number of spectators were able to remember the following snippets of information:

Only one person wore the same number as the position they finished. Gary, who didn't wear green, beat Barry. Larry beat the person who wore yellow. The person who wore number 3, wore green. The person who wore number 2 finished first whereas Harry came last. The person who finished second wore green, Barry wore yellow and the person wearing red beat the person wearing blue.
Can you work out who finished where, the number and colour they wore?
 

Answer:

 # Name   Wore  Colour
 1 Gary   2     red
 2 Larry  3     green
 3 Barry  1     yellow
 4 Harry  4     blue

Plane and Train Spotting Contest

During a recent plane and train spotting contest, five eager entrants were lined up ready to be tested on their spotting ability. They had each spotted a number of planes (26, 86, 123, 174, 250) and a number of trains (5, 42, 45, 98, 105). From the clues below, can you determine what colour anorak each was wearing, their position, their age (21, 23, 31, 36, 40) and the number of trains and planes spotted?

 1. Simon spotted 44 fewer trains than planes.
 2. Keith was 36 years old.
 3. The person on the far right was 8 years younger than Simon, and spotted 174 planes.
 4. James was wearing a beige anorak and spotted 37 trains fewer than Simon.
 5. The person who was wearing a green anorak, was 19 years younger than the person to the left of him.
 6. Steven spotted 105 trains and 250 planes.
 7. The person in the centre was 31 years old, was wearing a blue anorak and spotted 42 trains.
 8. Alan, who was on the far left, spotted 26 planes, and spotted 72 trains more than planes.
 9. The person who was wearing a red anorak, was 4 years older than Keith and was not next to the person wearing a blue anorak.
 10.The person who was next to the 31 year old but not next to the person who spotted 26 planes, was wearing a orange anorak, and spotted 45 trains.

Answer:

# Name    Anorak  Age  Planes  Trains
 1 Alan    red     40    26      98
 2 Steven  green   21   250     105
 3 Simon   blue    31    86      42
 4 Keith   orange  36   123      45
 5 James   beige   23   174       5

 

Word Odd Out

Which of the following words is odd one out?

1. actualized
2. authorizer
3. customized
4. carbonized
5. scandalize

Answer:

No. 3. All other words share 4 letters A, E, I, and Z.

PRNNGSLWDRSCP!

Four words related to basic gardening have been written in a row with neither spaces nor vowels. What are these four words?

PRNNGSLWDRSCP

Answer:

Pruning Soil Weeder Scoop

Which is the odd one out?

Which is the odd one out?

Answer:

C.

C has three white and one black on the left and three black and one white on the right. The others are the opposite way round.

X=? Y=?

Two numbers, x and y, are such that if x gets 77 from y they will be in the ratio 4 to 1, but if y receives 42 from x, their ratio will be 1 to 2. What are x and y?  

Answer:

x=127 and y=128

Which tile is missing?

Which tile is missing?

Answer:

In each row and column, every ring is shaded black once.

Missing Number

1. What is the missing number?

14
4 7 4
5 1 4 1 5
1 2 3 5 3 2 1
1 1 1 1 ? 1 1 1 1

a. 1
b. 2
c. 6
d. 10

Answer:

d

14 = 14
4 + 7 + 4 = 15
5 + 1 + 4 + 1 + 5 = 16
1 + 2 + 3 + 5 + 3 + 2 + 1 = 17
1 + 1 + 1 + 1 + 10 + 1 + 1 + 1 + 1 = 18

Numbers in Shapes!

In the following figure, identify the values of x and y such that:

1) The sum of numbers inside the circle must be individually higher than the heart, the diamond and the triangle.

2) The sum of numbers inside the rectangle must be half of the total of the sum of numbers inside the circle, the heart, the diamond and the triangle.

3) Both x and y should be divisible by 5.

Answer:
x = 30
y = 25

Who stole the cake?

During a recent police investigation, Chief Inspector Stone was interviewing five local villains to try and identify who stole Mrs Archer's cake from the mid-summers fayre. Below is a summary of their statements:

Arnold:  it wasn't Edward
          it was Brian
 Brian:   it wasn't Charlie
          it wasn't Edward
 Charlie: it was Edward
          it wasn't Arnold
 Derek:   it was Charlie
          it was Brian
 Edward:  it was Derek
          it wasn't Arnold

It was well known that each suspect told exactly one lie. Can you determine who stole the cake?

 

Answer:

Charlie committed the terrible crime.
Looking at Brian's statements, one of the statements was a lie and the other was the truth. Therefore it must have been either Charlie or Edward.
Looking at Derek's statements, for the same reason, it was either Charlie or Brian.
Therefore it must have been Charlie who committed the crime. Double checking this against the other statements confirms this.

 

Trace out a 13-letter word!

In this figure, trace out a 13-letter word by travelling along the lines without crossing a letter twice.

Answer:
Cachinnation  (cachinnation means "loud convulsive laughter")

O = Odd, E = Even

In the following sum the digits 0 to 9 have all been used, O = Odd, E = Even, zero is even and the top row's digits add to 9. Can you determine each digit?

 

 

Answer:

Remembering that:
    E + E = E
    O + O = E
    E + O = O
To discuss individual letters it's easiest to represent the sum as:
    A B C
    D E F +
 --------
  G H I J
The largest values for A and D are 6 and 8, which makes G = 1.
 Since column 2 is E + O = O there can be no carry from column 1 (since E + O + 1 is always even). Therefore C and F are 3 and 5 (but we don't yet know which is which), therefore J = 8.
 There can't be a carry from column 2 (as A + D is even) therefore E can't be 9 as this would force a carry.
 Therefore I = 9. Hence B can't be 0. Therefore H = 0.
 The last remaining odd number makes E = 7. Making B = 2.
 Therefore A and D are 4 and 6 (but we don't yet know which is which).
 Since the top row's digits have to add to 9 the top number must be 423.
 This makes the sum 423 + 675 = 1098.
 

Rooms

Alex, Bret, Chris, Derek, Eddie, Fred, Greg, Harold, and John are nine students who live in a three storey building, with three rooms on each floor. A room in the West wing, one in the centre, and one in the East wing. If you look directly at the building, the left side is West and the right side is East. Each student is assigned exactly one room. Can you find where each of their rooms is:

1. Harold does not live on the bottom floor.
2. Fred lives directly above John and directly next to Bret (who lives in the West wing).
3. Eddie lives in the East wing and one floor higher than Fred.
4. Derek lives directly above Fred.
5. Greg lives directly above Chris.
 

Answer:

From the highest floor to lowest we have:
West    Centre  East
====    ======  ====
Harold  Derek   Eddie
Bret    Fred    Greg
Alex    John    Chris

Back to Starting Position

I have a machine which has four sequential cog wheels in constant mesh.

The largest cog has 81 teeth and the others have 52, 36 and 20 respectively.

What is the fewest number of revolutions the largest cog must make so that all of the cogs are back in their starting position?

 

 

Answer:

260 revolutions.

There are a number of ways of thinking about the solution, and we find this one the quickest way to find the answer.
The total number of teeth moved by Cog 1 will be wholly divisible by each cog in turn, therefore:

Revolutions x Cog 1 ÷ Cog2 is an integer
Revolutions x Cog 1 ÷ Cog3 is an integer
Revolutions x Cog 1 ÷ Cog4 is an integer

So we are after the first number of revolutions x 81 that is an integer after division by 52, 36 and 20.
Thus:
 81     81     81
 -- and -- and -- all need to be integers (and not fractions).
 52     36     20
 An easy way to do this would be to multiply by 52 x 36 x 20 = 37,440 revolutions, which would be a correct answer, but not necessarily the smallest answer.
 A better way is to break each cog down into its prime factors, where Cog 1 has the largest number of teeth:

 Cog 1 - 81 = 3 x 3 x 3 x 3
 Cog 2 - 52 = 2 x 2 x 13
 Cog 3 - 36 = 2 x 2 x 3 x 3
 Cog 4 - 20 = 2 x 2 x 5

 3 x 3 x 3 x 3 and 3 x 3 x 3 x 3 and 3 x 3 x 3 x 3 and these need to be integers
 -------------     -------------      ------------
 2 x 2 x 13       2 x 2 x 3 x 3       2 x 2 x 5
 Simplifying any fraction that can be simplified gives:
 3 x 3 x 3 x 3 and 3 x 3 and 3 x 3 x 3 x 3 and these need to be integers
 -------------     -----     -------------
 2 x 2 x 13       2 x 2       2 x 2 x 5
 Multiplying throughout by 2 x 2 gives:
 3 x 3 x 3 x 3 and 3 x 3 and 3 x 3 x 3 x 3 and these need to be integers
 -------------               -------------
     13                            5
 Multiplying throughout 13 gives:
 3 x 3 x 3 x 3 and 3 x 3 x 13 and 3 x 3 x 3 x 3 x 13 and these need to be integers
                                  ------------------
                                         5
 Multiplying throughout 5 gives:
 3 x 3 x 3 x 3 x 5 and 3 x 3 x 13 x 5 and 3 x 3 x 3 x 3 x 13
 They are all now integers, and we have therefore multiplied by 2 x 2 x 13 x 5.
 2 x 2 x 13 x 5 = 260 revolutions. As required.
 The easy way from above of 37,440 is exactly 144 times this answer.