0
votes
0answers
35 views

Why is it true that $\forall b\in(0,1): (1-b)\left(e(1-b)\right)^{\frac{b}{1-b}}\geq\prod\limits_{n=2}^{\infty}n^{-b^n}\geq 0$

Why is it true that $$\forall b\in(0,1)$$ $$1\geq(1-b)\left(e(1-b)\right)^{\frac{b}{1-b}}\geq\prod\limits_{n=2}^{\infty}n^{-b^n}\geq 0$$ Note: Let $$f(x)=\prod\limits_{n=2}^{\infty}n^{-b^n}$$ Then ...
2
votes
3answers
48 views

Why is this true? $\forall a\in(1,\infty), B\in(0,\infty), x\in(0,\infty) : a^x\geq \left(\frac{ex\ln(a)}{B}\right)^{B}$

I know $$\forall a\in(1,\infty), B\in(0,\infty), x\in(0,\infty)$$ $$a^x\geq \left(\frac{ex\ln(a)}{B}\right)^{B}$$ can be proved using AM-GM. Is there a simple way to show the inequality holds in all ...
4
votes
2answers
80 views

What comes next? For 8 year olds. Part II

This question is from the homework of my niece. She is 8 years old. And I could not help her with this question. There is a sequence of numbers. Problem asks the sum of the next two numbers. And ...
0
votes
1answer
75 views

What comes next? For 8 year olds

This question is from the homework of my niece. She is 8 years old. And I could not help her with this question. There are 5 x 3 cells. And there is a number in each cell. Problem asks what should be ...
57
votes
5answers
9k views

Help me solve my father's riddle and get my book back

My father is a mathteacher and as such he regards asking tricky questions and playing mathematical pranks on me once in a while as part of his parental duty. So today before leaving home he sneaked ...
13
votes
6answers
2k views

A strange puzzle having two possible solutions

A friend of mine asked me the following question: Suppose you have a basket in which there is a coin. The coin is marked with the number one. At noon less one minute, someone takes the coin ...
0
votes
2answers
112 views

What comes next in the series?

Well, here is a numerical sequence puzzle, which I really tried hard, but still could not find the pattern? 2 4 6 30 32 34 36 40 42 44 46 50 52 54 56 60 62 64 66 x? Can anybody help me out in ...
3
votes
1answer
210 views

Apparent paradox for the bird traveling between two trains puzzle

Gretings. Trying the "hard solution" for the puzzle below (which has been discussed, with a different angle, elsewhere on this forum) I got to a point where I have three seemingly valid solutions, ...
29
votes
4answers
567 views

Fibonacci numbers from $998999$

Is there a nice explanation of ...
1
vote
0answers
66 views

Math Puzzle, finding a sequence with a certain property

A certain number of the $5000$ members of the Conway Mathematical Society (each of which has a different membership number between $1$ and $5000$) got together to discuss a puzzle. Much to their ...
0
votes
2answers
83 views

Riddle about ant going through self extendible string

Ant stands at the end of a rubber string which has 1km of length. Ant starts going to the other end at speed 1cm/s. Every second the string becomes 1km longer. For readers from countries where ...
4
votes
1answer
54 views

Constructing of sequences with steps already “existing” in sequence

Sequence $\{a_k\}$ is built according to next rule: $$ a_0=0;\\ a_1=1;\\ \forall n\in \mathbb{N} ~ \exists i,j ~ (0\leqslant i<j\leqslant n),\mbox{ so that } a_{n+1}-a_{n} = a_j-a_i. $$ ...
1
vote
3answers
149 views

What is the next number? [closed]

What is the next number in the following set ? $$1,11,21,1211,111221, \ldots$$
3
votes
0answers
170 views

Limits of infinite processes that terminate in finite time - checking my understanding?

I am a computer scientist by training, but have a fair amount of math background that I've picked up through classes, teaching, and general interest. A student of mine posed a question to me. I think ...
2
votes
1answer
183 views

roulette wheel sequence

Is the sequence of numbers around a European roulette wheel (the integers from 0 to 36 inclusive) random or is there a pattern to it? It is said to have been devised by Pascal, which might be thought ...
0
votes
1answer
130 views

Pascal's other triangle

Just a brainteaser question: Can you identify the generator of the following pattern of numbers?      Remark on any interesting patterns you see in the triangle.
49
votes
3answers
16k views

Predicting Real Numbers

Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to ...
0
votes
1answer
45 views

Find out the length of a recurrence

I have this rules for creating a list of numbers: $x/2$ if $x$ is even, repeat $3x+1$ if $x$ is odd, repeat if $x=1$, stop so for example, starting from 15, the list will be: 15, 46, 23, 70, 45, ...
4
votes
1answer
177 views

What function does this infinite series represent?

$$\frac14+\frac{x-4}{2!x^2}-\frac{(x-4)(2x-4)(3x-4)}{4!x^4}+\frac{(x-4)(2x-4)(3x-4)(4x-4)(5x-4)}{6!x^6}\mp\ldots$$ Can anyone deduce the sum of this series? The reason I ask is because I made it and ...
14
votes
4answers
6k views

How is this a number sequence 58, 26, 16, 14, 10

I recently had a IQ Test taken and we all got stuck on the same question. The question was: What comes next in the following sequence? 58, 26, 16, 14, _ The answer given in the answer sheet was 10. ...
2
votes
1answer
86 views

About some finite sequences of integers

The following sequence of p = 7 terms: 5 ; -3 ; 1 ; -4 ; 6 ; -4 ; 1 has a positive sum, and each sum of q = 4 consecutive terms is negative. Does anybody know the general conditions on p and q to ...
0
votes
0answers
60 views

What general function or rounding method can be derived from these number series?

OK, first of all, don't laugh, this is related to a social iOS game, but I promise there is some real juicy math here... I am attempting to derive a general formula or algorithm that can predict the ...
5
votes
1answer
1k views

Fly and Two Trains Riddle

Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 100km apart. A fly starts at the front of one train and flies at 75 kph to the front of the ...
3
votes
2answers
309 views

puzzle about array of numbers

Consider an array of numbers $$ \color{#C00000}{1}\ \hphantom{7\ 6\ 5\ 4\ 7\ 3\ 5\ 7\ 2\ 7\ 5\ 3\ 7\ 4\ 5\ 6\ 7\ }\color{#C00000}{1}\\ 1\ \hphantom{7\ 6\ 5\ 4\ 7\ 3\ 5\ 7\ }\color{#C00000}{2}\ ...
2
votes
6answers
529 views

Next number in series

What are the basic/advanced strategies used to find the next number in series. I know the simple ones such as addition, multiplication etc. But recently I came into a question that goes on something ...
27
votes
1answer
665 views

A fun Pascal-like triangle

Inspired by Pascal, I put on some shackles and a thorny belt. Inspiration came pouring in, and I thought of the following triangle: $$ \begin{array}{rcccccccccc} & & & & ...
1
vote
2answers
211 views

Zeno-like riddle with additional complication: Runner with dog running back and forth at different speeds

I came across this little riddle: An amateur runner trains on a 1 250 m long track at a speed of $v_L = 2 m/s$. His dog runs ahead to the target, turns around and then runs back to him, then he ...
1
vote
3answers
220 views

Sum the infinite series $ \sum_{n=0}^\infty (2n^7 + n^6 + n^5 + 2n^2)/n! $

What is $$ \sum_{n=0}^\infty \frac{2n^7 + n^6 + n^5 + 2n^2}{n!} $$
29
votes
3answers
1k views

Tall fraction puzzle

I was given this problem 30 years ago by a coworker, posted it 15 years ago to rec.puzzles, and got a solution from Barry Wolk, but have never seen it again. Consider the series: $$1, ...
3
votes
2answers
4k views

Famous puzzle: Girl/Boy proportion problem (Sum of infinite serie)

Puzzle In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is ...
4
votes
6answers
768 views

A tedious puzzle (but not homework)

There are 1000 light bulbs and 1000 tutors. All light bulbs are off. Tutor 1 goes flipping light bulb 1,2,3,4... tutor 2 then flips 2,4,6,8...tutor 3 then 3,6,9...etc until all 1000 tutors have done ...