Number Puzzles and Sequences

Easy Deduction

A teacher thinks of two consecutive numbers between 1 and 10. The first student knows one number and the second student knows the second number. The following exchange takes place:
First: I do not know your number.
Second: Neither do I know your number.
First: Now I know.
What are the 4 solutions of this easy number puzzle?

Complex Deduction

This is definitely one of the harder number puzzles on this site.
A teacher says: I'm thinking of two natural numbers greater than 1. Try to guess what they are.
The first student knows their product and the other one knows their sum.
First: I do not know the sum.
Second: I knew that. The sum is less than 14.
First: I knew that. However, now I know the numbers.
Second: And so do I.
What were the numbers?

Children

An easier number puzzle is as follows. Two friends are chatting:
- Peter, how old are your children?
- Well Thomas, there are three of them and the product of their ages is 36.
- That is not enough ...
- The sum of their ages is exactly the number of beers we have drunk today.
- That is still not enough.
- OK, the last thing is that my oldest child wears a green cap.
How old were each of Peter's children?

Symbol

What mathematical symbol can be placed between 5 and 9, to get a number greater than 5 and smaller than 9?

Fraction

Can you arrange 9 numerals - 1, 2, 3, 4, 5, 6, 7, 8 and 9 - (using each numeral just once) above and below a division line, to create a fraction equaling to 1/3 (one third)?

5-digit Number

What 5-digit number has the following property? If we put numeral 1 in front of the number, we get a number three times smaller, than if we put the numeral 1 behind this number.

9-digit Number

Find a 9-digit number, which you will gradually round off starting with units, then tenth, hundred etc., until you get to the last numeral, which you do not round off. The rounding alternates (up, down, up ...). After rounding off 8 times, the final number is 500000000. The original number is commensurable by 6 and 7, all the numbers from 1 to 9 are used, and after rounding four times the sum of the not rounded numerals equals 24.

10-digit Number

• Find a 10-digit number, where the first figure defines the count of zeros in this number, the second figure the count of numeral 1 in this number etc. At the end the tenth numeral character expresses the count of the numeral 9 in this number.
• Find a 10-digit number, where the first numeral character expresses the count of numeral 1 in this number, ..., the tenth numeral the count of zeros in this number.

Cipher

Find the number if:

1. The cipher is made of 6 different numerals.
2. Even and odd digits alternate, including zero (in this case as an even number).
3. The difference between two adjacent numerals is always greater than one (in absolute value).
4. The first two numerals (as one number) as well as the two middle numerals (as one number) are a multiple of the last two numerals (as one number).

What is the cipher? There is more than 1 solution.

The Number Puzzle

The grid below is to be filled with six numbers (3 vertical and 3 horizontal) from the given list. You can use each number more than once. After completing the number puzzle sum up all digits in the grid. This is defined as the score. What is the maximum possible score?

The score is 73 (of course other solutions with higher scores are possible).

This one is not verified - I used 2 numbers 39543 and 89398. And this is what the grid looks like:

So the total score is 147.

Further Discussion

Master Mind

Find the four-digit number designated by asterisks, given the following:

• All four digits of the unknown number are different.
• None of the digits is zero.
• Each "0" on the right of each four-digit number below indicates that the number has a matching digit in a non-matching position with the unknown number.
• Each "+" on the right of each four-digit number below indicates that the number has a matching digit in a matching position with the unknown number.

              6152    +0
              4182    00
              5314    00
              5789     +
              -------------
              ****                        
              

1996

Using the numerals 1, 9, 9 and 6, mathematical symbols +, -, x, :, root and brackets create the following numbers:
29, 32, 35, 38, 70, 73, 76, 77, 100 and 1000.
All the numerals must be used in the given order (each just once) and without turning upside down.

I used [brackets] as the symbol for root.

Further Discussion

100

Using four sevens (7) and a one (1) create the number 100. Except the five numerals you can use the usual mathematical operations (+, -, x, :), root and brackets ().

Equation

Rectify the following equality 101 - 102 = 1 by moving just one digit.

Number Sequences

There are infinite formulas that will fit any finite series. Try to guess the following number in each sequence (using the most simple mathematical operations, because as I mentioned, there is more than one solution for each number sequence).

• 8723, 3872, 2387, ?
• 1, 4, 9, 18, 35, ?
• 23, 45, 89, 177, ?
• 7, 5, 8, 4, 9, 3, ?
• 11, 19, 14, 22, 17, 25, ?
• 3, 8, 15, 24, 35, ?
• 2, 4, 5, 10, 12, 24, 27, ?
• 1, 3, 4, 7, 11, 18, ?
• 99, 92, 86, 81, 77, ?
• 0, 4, 2, 6, 4, 8, ?
• 1, 2, 2, 4, 8, 11, 33, ?
• 1, 2, 6, 24, 120, ?
• 1, 2, 3, 6, 11, 20, 37, ?
• 5, 7, 12, 19, 31, 50, ?
• 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, ?
• 126, 63, 190, 95, 286, 143, 430, 215, 646, 323, 970, ?
• 4, 7, 15, 29, 59, 117, ?
• 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, ?
• 4, 4, 341, 6, 4, 4, 6, 6, 4, 4, 6, 10, 4, 4, 14, 6, 4, 4, 6, 6, 4, 4, 6, 22, 4, 4, 9, 6, ?