The worlds first riddle!

Math free zone

What is the next number in the sequence - 77, 49, 36, 18,

A boat will carry only 200 pounds. How can a man weighing 200 pounds. His two sons, each weighing 100 pounds and a sack of potatoes weighing 100 pounds use the boat to cross the river

A bottle and a cork together cost $1.50. The bottle costs a dollar more than the cork. How much does each cost

Take two apples from three apples and what have you got



Math zone

Five men crash-land their airplane on a deserted island in the South Pacific. On their first day, they gather as many pineapples as they can find into one big pile. They decide that, since it is getting dark, they will wait until the next day to divide the pineapples.

That night each man took a turn watching for rescue searchers while the others slept. The first watcher got bored so he decided to divide the pineapples into five equal piles. When he did this, he found he had one remaining pineapple. which he gave to a monkey, took one of the piles, and hid it for himself. Then he jumbled up the four other piles into one big pile again.

To cut a long story short, each of the five men ended up doing exactly the same thing. They each divided the pineapples into five equal piles and had one extra pineapple left over, which they gave to the monkey. They each took one of the five piles and hid those pineapples. They each came back and jumbled up the remaining four piles into one big pile.

What is the smallest number of pineapples there could have been in the original pile

[size=7]1) 11
2) 1st over 1boy 1sack potatoes, 1st back 1 boy, 2nd over 2 boys, 2nd back 1 boy, 3rd over man, 3rd back boy, 4th over 2 boys.
3) X+Y=1.50, X-Y=1.00 so 2X=2.50 & X=1.25
bottle cost 1.25 and cork =0.25
4) Two apples
5) is there an extra pineapple left over in the morning? [/size]
Rap

[size=7]NEXT NUMBER
8 (or 08)

BOTTLE/CORK
bottle = $1.25, cork = $.25

APPLES
2

PINEAPPLES
3121[/size]

next number = 8

son 1 and sack cross
son 1 returns
son 1 and 2 cross
son 2 returns
man crosses
son 1 returns
son 1 and 2 cross

Apples = 2

bottle= 1.25 and cork = .25 [size=7][/size]

Mark:

LADDERS
6 (similar triangles: x/4 = (9+x)/(4+x))


By similar triangles, 9/h = h/4.
Hence h2 = 36.

Rejecting the negative root, h, the vertical height of the intersection above the ground, is 6 feet.


COINS
77 Mark, you make it look so easy.

Suppose we have x coins, comprised of: a pennies, b nickels, c dimes, d quarters, e half dollars, and f dollars, with a, b, c, d, e, f 0.

Then a + 5b + 10c + 25d + 50e + 100f = 100 and x = a + b + c + d + e + f.

Therefore 4b + 9c + 24d + 49e + 99f = 100 - x.

We seek the smallest value of x such that the above equation has no solution.
In other words, we seek the largest value of 100 - x with no solution.
That is, the largest value that cannot be obtained by substituting non-negative integers into 4b + 9c + 24d + 49e + 99f.

This is not as intractable, and it's quite easy to verify that 23 is the highest such value. In particular, this approach simplifies the checking process for small values of x, i.e., large numbers of coins.

Therefore the smallest number of coins with which it is impossible to make a dollar is 77.





1. It depends on the definition of "meet." The train that left NYC will be closer to NYC until the front of the other train passes its rear.

Good point, the question needs reworking.

2. 20

3. today (Monday)

4. Thursday the 17th

5. Does 97% caffeine-free mean that the coffee is 3% caffeine or has 3% of the regular amount of caffeine? If the former, the amount of caffeine in a regular cup would need to be known. If the latter, 33 1/3 cups.

The latter: Because there is 3 % caffeine left in the doctored coffee; in 100 cups there would be enough for 3 cups of regular; 3 goes into 100 exactly 33 1/3.


NEXT NUMBER
8 (or 08)

Why? 7x7=49, 4x9=36, 3x6=18, 1x8=8




BOTTLE/CORK
bottle = $1.25, cork = $.25

APPLES
2

PINEAPPLES
3121


Let the original pile have n pineapples. Let a be the number of pineapples in each of the five piles made by the first man, b the number of pineapples in each of the five piles made by the second man, and so on.
Writing a Diophantine equation to represent the actions of each man, we have

n = 5a + 1<> n + 4 = 5(a + 1)
4a = 5b + 1<> 4(a + 1) = 5(b + 1)
4b = 5c + 1<> 4(b + 1) = 5(c + 1)
4c = 5d + 1<> 4(c + 1) = 5(d + 1)
4d = 5e + 1<> 4(d + 1) = 5(e + 1)

Hence n + 4 = 5 × (5/4)4 (e + 1), and so n = (55/44) (e + 1) - 4.

Note that, since 5 and 4 are coprime, 55/44 = 3125/256 is a fraction in its lowest terms. Hence the only integer solutions of the above equation are where e + 1 is a multiple of 44, whereupon d + 1, c + 1, b + 1, and a + 1 are all integers.

So the general solution is n = 3125r - 4, where r is a positive integer, giving a smallest solution of 3121 pineapples in the original pile.


Rap:

1) 11 (See above)

2) 1st over 1boy 1sack potatoes, 1st back 1 boy, 2nd over 2 boys, 2nd back 1 boy, 3rd over man, 3rd back boy, 4th over 2 boys.

3) X+Y=1.50, X-Y=1.00 so 2X=2.50 & X=1.25
bottle cost 1.25 and cork =0.25

4) Two apples

5) is there an extra pineapple left over in the morning? (Ask the minkey)





Each child in a family has at least three brothers and two sisters. What is the smallest number of children the family might have



A number of children are standing in a circle. They are evenly spaced and the fourth child is directly opposite the twentieth child. How many children are there altogether



Two players bet on the total roll of two standard dice. Player A bets that a 12 will be rolled first. Player B bets that two consecutive 7s will be rolled first. The players keep rolling until one player wins.

What is the probability that A will win



A confused bank teller transposed the dollars and cents when he cashed a check for Ms Jones, giving her dollars instead of cents and cents instead of dollars. After buying a newspaper for 50 cents, Ms Jones noticed that she had left exactly three times as much as the original check.

What was the amount of the check


Wot! Is that it? No more numbers totals or change. You would have to be Captain Marvel to come up with an answer.

G'day Dadpad, sorry to have missed you, but it takes a long time to get the replies in order and new questions sorted. Anyhoo:

next number = 8

son 1 and sack cross
son 1 returns
son 1 and 2 cross
son 2 returns
man crosses
son 1 returns
son 1 and 2 cross

Apples = 2 Although I like a nice pear myself!

bottle= 1.25 and cork = .25

I could not have done better myself. I'll have that beer now!

[size=7]FAMILY
7 (4 boys, 3 girls)

CIRCLE
32

DICE
Still thinking. .5 seems obvious, but feels wrong

CHECK
$18.56[/size]

[size=7]DICE
In a trial of 100,000,000 games, A won 53,786,412 times. So, I'll say 53.8% without any reasoning to back it up.[/size]

What do the following represent

A) 1001 = A.N.
B) 54 = C. in the D. (with the J.'s)
C) 29 = D. in F. in L.Y.
D) 24 = H. in a D.
E) 13 = S. on the A.F.
F) 26 = L. of the A.


If factorial 100 (100!) was rewritten as the product of its prime factors how many 2's and how many 5's would there be

1001 Arabian Nights
29 Days in February in Leap Years
24 Hours in a Day
13 Stripes on the American Flag
26 Letters of the Alphabet (or 30 = S. u A. --- 30 Slova u Abecedi ).


and

54 Costaricans in the Dugout (with the Jagermeisters)

[size=7]54 = C. in the D. (with the J.'s)
cards, deck, jokers[/size]

Is that 100 * 100! or 100^100!?

Mark:


FAMILY
7 (4 boys, 3 girls)

CIRCLE
32

The number of children is 2 x difference between the opposite numbers, since counting the opposite numbers gets you half way round the circle (think of it in terms of a clock if this helps, if not just ask Mark), so there are 32 children.



CHECK
$18.56 .


Let x be the number of dollars in the check, and y be the number of cents.
Then 100y + x - 50 = 3(100x + y).
Therefore 97y - 299x = 50.

A standard solution to this type of Diophantine equation uses the Euclidean algorithm.
The steps of the Euclidean algorithm for calculating the greatest common divisor (gcd) of 97 and 299 are as follows:

299 = 3 × 97 + 8
97 = 12 × 8 + 1

This shows that gcd (97,299) = 1.

To solve 97y - 299x = gcd (97,299) = 1, we can proceed backwards, retracing the steps of the algorithm as follows:

1 = 97 - 8 × 12
= 97 - (299 - 3 × 97) × 12
= 37 × 97 - 12 × 299

Therefore a solution to 97y - 299x = 1 is y = 37, x = 12.
Hence a solution to 97y - 299x = 50 is y = 50 × 37 = 1850, x = 50 × 12 = 600.

It can be shown that all integer solutions of 97y - 299x = 50 are of the form y = 1850 + 299k, x = 600 + 97k, where k is any integer.

In this case, because x and y must be between 0 and 99, we choose k = -6.
This gives y = 56, x = 18.
So the check was for $18.56.

Solution by Simultaneous Equations
Let x be the number of dollars in the check, and y be the number of cents.
Then, after buying the newspaper, Ms Jones has 3x + 3y, of which the cents portion is x - 50.

Equating dollars and cents, and depending upon the relative values of x and y, one of the following three sets of simultaneous equations must be true:

3x = y, 3y = x - 50.
3x = y - 1, 3y = x + 50.
3x = y - 2, 3y = x + 150.

Only the third set of equations has positive integer solutions, which are x = 18, y = 56.
So the check was for $18.56.



DICE
Still thinking. .5 seems obvious, but feels wrong

In a trial of 100,000,000 games, A won 53,786,412 times. So, I'll say 53.8% without any reasoning to back it up.

(I can think of 1000,000,000 reasons)

Let p be the probability that player A wins. Consider the following mutually exclusive cases, which encompass all possibilities.

If the first roll is a 12 (probability 1/36), A wins.
If the first roll is neither a 7 nor a 12 (probability 29/36), A wins with probability p.
If the first roll is a 7 and the second roll is a 12 (probability 1/6 • 1/36 = 1/216), A wins.
If the first roll is a 7 and the second roll is neither a 7 nor a 12 (probability 1/6 • 29/36 = 29/216), A wins with probability p.
If the first and second rolls are both 7 (probability 1/6 • 1/6 = 1/36), A cannot win.
Probability p is the weighted mean of these possibilities.
So p = 1/36 + (29/36)p + 1/216 + (29/216)p.
Therefore p = 7/13.



"Is that 100 ?"

That is: 100! = 100 x 99 x 98 x 97 x ... x 3 x 2 x 1


My O good to see U. How is your day job going? Any good ?'inside' stories?

1001 Arabian Nights
29 Days in February in Leap Years
24 Hours in a Day
13 Stripes on the American Flag
26 Letters of the Alphabet (or 30 = S. u A. --- 30 Slova u Abecedi


I also forgot the 3,000 + letters in the Chinese alphabet.



What's in seasons, seconds, centuries and minutes but not in decades, years, or days


One pencil can be made from four nubs of used pencils. If a man uses 16 pencils down to nubs, how many more pencils can he make


Amoebas reproduce by splitting in two. An amoeba that does so every minute is placed in a jar at exactly 10 o'clock in the morning. At 12:00 noon the jar is full. At what time is the jar half full


A prisoner is told "If you tell a lie we will hang you; if you tell the truth we will shoot you." What can he say to save himself



Four actuarial students sitting in their office decide to test the logic skills of their peers. The divide themselves into two pairs. One pair, of TRIOS, count in threes and the other pair, of SEPTOS, count in sevens (so a Trio writes 22 and a Septo writes 11 for the number eight).

They tell their colleagues a number which describes the worth of the furniture that they can see when they all stand in the centre of the room and face in different directions. A desk is worth 7 actuarial units and a chair is worth 3.

The person facing North says 41 units, East says 22 units, South says 101 units and West says 122 units.

How many desks and how many chairs are there altogether in the office

Actuarial problem

[size=7]Step 1
Putting responses into base 10
-------trios-----septos
41------13----------29
22-------8----------16
101-----10---------50
122-----17---------65

Step 2 finding numbers of possible desks and chairs
North
Trios 13=1*7+2*3 so d=1 c=2
Septos 29=2*7+5*3 so d=2 c=5

East
Trios 8---not possible
Septos 16=1*7+3*3 so d=1 c=3

South
Trios 10=1*7+1*3 so d=1 c=1
Septos 50=2*7+13*3=5*7=5*3 so d=(2 or 5) c=(12 or 5)

West
Trios 17=2*7+1*3 so d=2 c=1
Septos 65=2*7+17*3=5*7+10*3 so d=(2 or 5) c=(17 or 10)

Step 3 finding corners
Corners (three possibilities) Septos can only be at East
Case----A---------B----------C
North-Trios-----Trios----Septos
East---Septos---Septos---Septos
South-Trios-----Septos---Trios
West--Septos---Trios-----Trios

Step 4 potential solutions
Case A d=1n+1e+1s+2/5w=5/8 c=2n+3e+3s+17/10w=23/16
Case B d=1e+1e+2/5s+2w=6/9 c=2n+3e+12/5s+1w=18/13
Case C d=2n+1e+1s+2w=6 c=5n+3e+1s+1w=8

Step 5 assumption and conclusion
Knowing how an actuarial likes unique solutions I'd choose
Case C
Septos facing North seeing 2 desk and 5 chairs (2*7+5*3=29 in base 7=41)
Septos facing East* seeing 1 desk and 3 chairs (1*7+3*3=16 in base 7=22)
Trios facing South seeing 1 desk and 1 chair (1*7+1*3=10 in base 2=101
Trios facing West seeing 2 desks and 1 chair (2*7+3=17=122)
* this is given in step 2[/size]
Rap

[size=7]Seasons, seconds, centuries and minutes---n

16 pencils-5

full-jar- 11:59

prisoner response - "You will shoot me."[/size]

Rap

[size=7]SEASONS, ETC.
n

PENCILS
5

AMOEBAS
11:59 a.m.

PRISONER
You'll hang me.

ACTUARIES
North and East are Septos (North because 41 isn't a valid base 3 n umber, East because 8 doesn't produce a valid desk/chair combination)
Therefore, South and West are Trios
After that, there is only one valid desk/chair combination per person.

North: 2 desks, 5 chairs
East: 1 desk, 3 chairs
South: 1 desk, 1 chair
West: 2 desks, 1 chair

Total: 6 desks, 10 chairs

100!
2: 97
5: 24[/size]

seasons seconds etc = N
Nubs= 5, 4 from first 16 and 1 from nubs resulting from those 4
amoebas= 11.59am (Slippery little suckers)

Rap:

(If they gave prizes for a beautifully crafted answer, then you would have them by the sack full.)


Step 1
Putting responses into base 10
-------trios-----septos
41------13----------29
22-------8----------16
101-----10---------50
122-----17---------65

Step 2 finding numbers of possible desks and chairs
North
Trios 13=1*7+2*3 so d=1 c=2
Septos 29=2*7+5*3 so d=2 c=5

East
Trios 8---not possible
Septos 16=1*7+3*3 so d=1 c=3

South
Trios 10=1*7+1*3 so d=1 c=1
Septos 50=2*7+13*3=5*7=5*3 so d=(2 or 5) c=(12 or 5)

West
Trios 17=2*7+1*3 so d=2 c=1
Septos 65=2*7+17*3=5*7+10*3 so d=(2 or 5) c=(17 or 10)

Step 3 finding corners
Corners (three possibilities) Septos can only be at East
Case----A---------B----------C
North-Trios-----Trios----Septos
East---Septos---Septos---Septos
South-Trios-----Septos---Trios
West--Septos---Trios-----Trios

Step 4 potential solutions
Case A d=1n+1e+1s+2/5w=5/8 c=2n+3e+3s+17/10w=23/16
Case B d=1e+1e+2/5s+2w=6/9 c=2n+3e+12/5s+1w=18/13
Case C d=2n+1e+1s+2w=6 c=5n+3e+1s+1w=8

Step 5 assumption and conclusion
Knowing how an actuarial likes unique solutions I'd choose
Case C
Septos facing North seeing 2 desk and 5 chairs (2*7+5*3=29 in base 7=41)
Septos facing East* seeing 1 desk and 3 chairs (1*7+3*3=16 in base 7=22)
Trios facing South seeing 1 desk and 1 chair (1*7+1*3=10 in base 2=101
Trios facing West seeing 2 desks and 1 chair (2*7+3=17=122)
* this is given in step 2





Answer:
We know that two of the students are counting in base 3 and two in base 7, but we don't know which ones. The answers in base 10 must also be divisible by a combination of 7 and 3.

Equivalent in base 10 if a Septo. Equivalent in base 10 if a Trio
North 41 29 n/a
East 22 16 8
South 101 50 10
West 122 65 17


North must therefore be able to see 2 desks (worth 7) and 5 chairs (worth 3). Noting that 8 is not divisible by any combination of 7 and 3, East must be a Septo and can see 1 desk and 3 chairs (totalling 16). South and West must therefore be Trios. South can therefore see 1 desk and 1 chair. West can see 2 desks and 1 chair.

Thus in total there are 6 desks and 10 chairs.



Rap:


Seasons, seconds, centuries and minutes---n

16 pencils-5

He makes four pencils from the sixteen nubs and then makes a fifth when he's done using those pencils.


full-jar- 11:59

prisoner response - "You will shoot me."


"You will hang me."


Mark:

SEASONS, ETC.
n

PENCILS
5

AMOEBAS
11:59 a.m.

PRISONER
You'll hang me.

ACTUARIES
North and East are Septos (North because 41 isn't a valid base 3 n umber, East because 8 doesn't produce a valid desk/chair combination)
Therefore, South and West are Trios
After that, there is only one valid desk/chair combination per person.

North: 2 desks, 5 chairs
East: 1 desk, 3 chairs
South: 1 desk, 1 chair
West: 2 desks, 1 chair

Total: 6 desks , 10 chairs

100!
2: 97
5: 24


The number of factors divisible by 2 is 50 (2 x 50 = 100)
The number of factors divisible by 4 is 25 (4 x 25 = 100)
The number of factors divisible by 8 is 12 (8 x 12 = 96)
The number of factors divisible by 16 is 6 (16 x 6 = 96)
The number of factors divisible by 32 is 3 (32 x 3 = 96)
The number of factors divisible by 64 is 1 (64 x 1 = 96)

So the number of 2s is 50+25+12+6+3+1 = 97

The number of factors divisible by 5 is 20 (5 x 20 = 100)
The number of factors divisible by 25 is 4( 25 x 4 = 100)
So the number of 5s is 20+4 = 24



Dadpad:

seasons seconds etc = N
Nubs= 5, 4 from first 16 and 1 from nubs resulting from those 4

Just in! From our scientific correspondent:

"amoebas (Slippery little suckers) = 11.59am




DP tells Amy that he has three cousins whose ages multiply to make 36 and add to make 13 and that the eldest cousin lives in Australia. What are the ages of DP's cousins


A 10-foot ladder hangs over the side of a boat with the bottom rung at the surface of the water. There is one foot between the rungs and the tide goes up at a rate of 6 inches per hour.

How long until three rungs are covered


How long will an eight-day clock run without winding



Jill offered Jack the following bet: she said she would toss three pennies in the air and if they fell all heads (H) or all tails (T) she would give him a dollar. If they fell any other way, he had to give her fifty cents.

Should Jack accept



Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cms can be fitted together, no gaps, or overlaps to form a rectangle.

What are the dimensions of the rectangle


Old /outstanding, does anyone want the answer for the two questions below?

What number comes next in this sequence:
7 8 5 5 3 4 4 ? 9

Two players take turns choosing one number at a time (without replacement) from the set {-4, -3, -2, -1, 0, 1, 2, 3, 4}. The first player to obtain three numbers (out of three, four, or five) which sum to 0 wins.

Does either player have a forced win?
(I have to say, "It don't look good")

Try:

I'd like to see the answer to the sequence problem. I had given up.

I'd like more time for the number selection problem. I had forgotten about it.

Thanks,
Mark

Mark, may I suggest you sit down before you read the answer. Also, please remove any sharp or pointy objects that may cause you to injure yourself, or me.


I am sure we are all familiar with Euler's famous formula for a polyhedron:

v - e + f = 2

Therefore if we study the problem as presented.

"What number comes next in this sequence:?"
7 8 5 5 3 4 4 ? 9


It immediately becomes apparent that the answer is ?'6'


Now, keep calm, take deep breaths.










Why? I hear you ask.

Well for a start…

It has NOTHING to do with Euler.












…August has only six (6) letters.





Keep up: January 7/February 8/ March 5/ etc…




Hello! Is that 911…..