#46

What number comes next in the sequence: 10, 9, 60, 90, 70, 66?

A) 96
B) 1000000000000000000000000000000000\
0000000000000000000000000000000000\
000000000000000000000000000000000
C) Either of the above
D) None of the above

Answer:

This can be looked up and found to be sequence A052196 http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A052196 in the On-Line Encyclopedia of Integer Sequences, which gives the largest positive integer whose English name has n letters. For example, the first few terms are ten, nine, sixty, ninety, seventy, sixty-six, ninety-six A more correct sequence might be ten, nine, sixty, googol, seventy, sixty-six, ninety-six, googolplex. And also note, incidentally, that the correct spelling of the mathematical term "googol" http://mathworld.wolfram.com/Googol.html differs from the name. The first few can be computed using the NumberName function in Eric Weisstein's MathWorld packages http://mathworld.wolfram.com/packages/

#45

1
1 1
2 1
1 2 1 1
1 1 1 2 2 1

What's the next line?

Answer:

312211.

This is the "look and say" sequence in which each term after the first describes the previous term: one 1 (11); two 1s (21); one 2 and one 1 (1211); one 1, one 2, and two 1's (111221); and so on. See the look and say sequence http://mathworld.wolfram.com/LookandSaySequence.html entry on MathWorld for a complete write-up and the algebraic form of a fascinating related quantity known as Conway's constant http://mathworld.wolfram.com/ConwaysConstant.html

#44

The following number is the only one of its kind.
8,549,176,320
Can you figure out what is so special about it?

Answer:

It's the only number that has all the digits arranged in alphabetical order.

#39

Consider the number 3 0 2 5. It satisfies several interesting properties:
It has four digits
All four digits are different

If we take the number that is the first two digits and sum it with the number that composes the last two digits, its square is the original number: (30+25)2=3025.
The original riddle asked you to find another such number. There is exactly one other, so altogether two. If the requirement that the digits be different is omitted, a third possibility presents itself. Finally, two more solutions enter into the game if we allow the four digits to include leading zeros, so all in all, five solutions to the relaxed version of this riddle.
It is this relaxed riddle that we want to consider. The riddle this month is composed of two questions. Answer both to be considered a solver.

Question 1:
If the relaxed version of the riddle is asked in an arbitrary base, b, instead of in base 10, how many solutions is it going to have, as a function of b?
Question 2:
Consider the strict (original) version of the riddle, but, once again, asked in base b instead of in base 10. Will there be an answer for every b≥4? Prove your claim.

I am still to solve this

#22

COMPLETE the following row of numbers: 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, 31, 100, 121, 10000, ?

Answer:

1 111 111 111 111 111
Yes, a very large number indeed! That's 16 ones in a row. And why? Well, think about the word 'COMPLETE' in the riddle. It means the row contains 16 numbers. So the row probably has something to do with the number 16.And how do these numbers relate to 16? Well, each number is actually equal to 16, but written in another numerical system. For example: 16 in the hexadecimal system is written as 10 (1x16 + 0x1). Let's go over the list:
16 in the 16th (=hexadecimal) numerical system is 10 (1 times 16 + 0 times 1)
16 in the 15th numerical system is 11 (1 times 15 + 1 times 1)
...
16 in the 3rd numerical system is 121 (1 times 9 + 2 times 3 + 1 times 1)
16 in the 2nd (binary) numerical system is 10000 (1 times 16 + 0 times 8 + 0 times 4 + 0 times 2 + 0 times 1)
and finally 16 in the 1st numerical system is 1111111111111111 (1 times 1 + 1 times 1 + ... + 1 times 1)

#10

There were 2 jugs, one could hold only 3 gallons of water and the other could hold only 5 gallons of water. How would you get exactly four gallons of water?

Answer:

You would put three gallons of water in the 5 gallon jug, then you would fill up the 3 gallon jug up, and pour as much water as you can into the 5 gallon jug. You will now have 1 gallon of water in the 3 gallon jug. You then empty the 5 gallon jug, put the one gallon of water from the 3 gallon jug into the 5 gallon jug then fill the 3 gallon jug up again and put the water into the 5 gallon jug.

#5

There is a 10 digit number, the digit 'n' in position 'p' tells the number of digits of p in that number. What is that number?

Answer:

position -> 0123456789
number -> 6210001000
6 is the number of 0s; 2 is the number of 1s; 1 is the number of 2s and 6s