Challenge Riddles and Puzzles: Extend the following to a valid equation

[fresh_42]

94. If $(F_n)_n$ is the ordinary Fibonacci sequence ($F_1=F_2=1$). What is
$$
\sum_{n=1}^\infty F_n \,10^{-(n+1)}
$$

D104

 

[Not anonymous]

Spoiler

$9\;9\;9\;=\;6$
If cube-root and square-root operations are acceptable, then here is one set of solutions:
$(9\; /\; \sqrt 9) + \sqrt 9=\;6$
$\sqrt[3]8 + \sqrt[3]8 + \sqrt[3]8 =\;6$
$7 - (7\; /\; 7) =\;6$
$(6\; / \; 6) \times 6 =\;6$
$(5\; /\; 5) + 5 =\;6$

 

[KnotTheorist]

Spoiler: Partial Spoiler to 94

We know that the sum converges by the ratio test. If we take the limit as n goes to infinity of |a_n+1/a_n|, we get F_n+110^-(n+2) / F_n10^-(n+1) = F_n+1/10F_n = φ/10 ≈.1618 < 1. Thus the summation converges.

From here, I am lost. I have done some calculations and written out the first hundred terms and they get close to .011235955056.

 

[bluej]

Spoiler: #94

$\sum_{n=1}^\infty F_n \,10^{-(n+1)} = \frac {1} {10} \sum_{n=1}^\infty \frac { F_n } {10^n}=\frac {1} {10} \frac {10} {89}=\frac {1} {89}=0.011235955$

 

[lpetrich]

Solution of #94

Spoiler

The Fibonacci series has closed-form solution
$$ F_n = \frac{1}{\sqrt{5}} \left[ \left( \frac{1+\sqrt{5}}{2} \right)^n - \left( \frac{1-\sqrt{5}}{2} \right)^n \right] $$
The sum that we want to calculate I will write more generally as
$$ S(a) = \sum_{n=0}^\infty F_n a^{n+1} $$
Inserting the Fibonacci series gives
$$ S(a) = \frac{a}{\sqrt{5}} \sum_{n=0}^\infty \left[ \left( a \frac{1+\sqrt{5}}{2} \right)^n - \left( a \frac{1-\sqrt{5}}{2} \right)^n \right] $$
We have geometric series here, so we get sums
$$ S(a) = \frac{a}{\sqrt{5}} \left[ \frac{1}{1 - a \frac{1+\sqrt{5}}{2}} - \frac{1}{1 - a \frac{1-\sqrt{5}}{2}} \right] = \frac{a^2}{1 - a - a^2} $$
Substituting in a = 1/10 gives
$$ S(1/10) = \sum_{n=1}^\infty F_n 10^{-n-1} = \frac{1}{89} $$

 

[fresh_42]

95. What are the smallest emirp and permutable three digit primes in the decimal system?

D104

 

[KnotTheorist]

Spoiler: 95 Spoiler

113
131
311

 

[fresh_42]

Spoiler: 95 Spoiler

113
131
311

Do you know the emirp, too?

 

[bluej]

Do you know the emirp, too?

Spoiler

smallest: 13; smallest 3-digit: 107

 

[fresh_42]

Spoiler

13

Three decimal digits!

 

[bluej]

Three decimal digits!

Knew you'd say dat added above in spoiler
Question not stated very precisely Mr 42..

 

[fresh_42]

Question not stated very precisely Mr 42..

... three digit primes ...

Question not read very precisely Mr. bluej..

 

[bluej]

Question not read very precisely Mr. bluej..

Can be read in 2 parts as 1) smallest emirp and 2) permutable three digit primes (i.e. all?)
Smallest 3-digit emirp in 10base system is 107.
Its more interesting in base 16!

 

[fresh_42]

Here's one I might regret ... for having to check the solutions ...

96. The rule is simple: choose a pin, jump over another horizontally or vertically into and if there is a hole, and remove the pin you jumped over. Is there a strategy which leads to a single pin in the middle of the board?
In order that we have a common notation, the first moves could (not have to) be:
1. 5-10-H / 10
2. 12-11-10 / 11
3 ...

D104

 

[mfb]

The Fibonacci sum without magic and without using the explicit formula:
Take out the first two elements:
$$\sum_{n=1}^\infty F_n 10^{-n-1} = 0.011 + \sum_{n=3}^\infty F_n 10^{-n-1}$$
Now use the recursive definition:
$$\sum_{n=3}^\infty F_n 10^{-n-1} = \sum_{n=3}^\infty (F_{n-2}+F_{n-1}) 10^{-n-1}$$
Some index shifting:$$= \sum_{n=1}^\infty F_{n} 10^{-n-3} + \sum_{n=2}^\infty F_{n} 10^{-n-2}$$
Let the second sum start at n=1 and subtract its first term, in addition unify the exponents:
$$= -0.001 + 0.01\sum_{n=1}^\infty F_{n} 10^{-n-1} + 0.1\sum_{n=1}^\infty F_{n} 10^{-n-1}$$
If we call the original sum X, then we can summarize this as $X = 0.011 - 0.001 + 0.01 X + 0.1 X$ or simplified $0.89 X = 0.01$, which is solved by $X=1/89$.

 

[fresh_42]

97. What is the smallest natural number $n$ such that $\varphi(n) \neq \varphi(p)$ for all primes $p\,?$

D105

 

[lpetrich]

#97 - presumably the Euler phi function, the count of all positive integers less than the arg that do not evenly divide the arg.

Spoiler

For prime number p, $\varphi(p) = p - 1$. Thus, for some n where there is no p that makes $\varphi(n)$ and $\varphi(p)$ equal, $\varphi(n) + 1$ must be a composite number. The smallest n that makes that happen is n = 15, and $\varphi(15) = 8$.

 

[fresh_42]

98. How many options are there to represent the number $1,000$ as sum of two or more consecutive natural numbers?

D106

 

[lpetrich]

#98

Spoiler

Let us first add the consecutive natural numbers between any two of them, $n_1$ and $n_2$. That sum is
$$ N = \sum_{n=n_1}^{n_2} n = \frac12 n_2(n_2 + 1) - \frac12 n_1(n_1 - 1) = \frac12 (n_2 - n_1 + 1) (n_2 + n_1) $$
or $2N = (n_2 - n_1 + 1) (n_2 + n_1)$. This expression means that a solution can be obtained by factoring 2N. Doing so gives $2N = f_1 f_2$ with the two f's being natural numbers, and one sets each factor equal to a factor of the sum expression: $f_1 = n_2 - n_1 + 1$ and $f_2 = n_1 + n_2$. Solving gives
$$ n_1 = \frac12 (f_2 - f_1 +1) ,\ n_2 = \frac12 (f_1 + f_2 - 1) $$
The pairs of lowest and highest natural numbers are thus

• 1000, 1000
• 198, 202
• 55, 70
• 28, 52

 

[fresh_42]

99. The regular $4,294,967,295-$gon can be constructed with compass and ruler. How?

D106

 

[fresh_42]

100. We are looking for an equilateral dodecagon whose sides form right angles at all vertices.

D106

 

[KnotTheorist]

Do you know the emirp, too?

My first guess at the problem was 107, but that can be permuted to 170 and 710, which are not prime.
I read the question as the smallest number that was both emirp and permutably prime.

 

[KnotTheorist]

Spoiler: 96

I do not have not the time nor the patience to write out the move set for completing this puzzle, even though it is one of my favorites. I own my own board and love to see people struggle with the challenge.
The penultimate move (of a fair and winning game) will always leave you in (5,10), (15, 16), (17, 18), or (23, 28). Thus you can either end the game with 1 peg in the center or on the respective edge.

 

[KnotTheorist]

Spoiler: spoiler 100

A plus sign, just like the logo of the American red cross

 

[KnotTheorist]

Spoiler: spoiler 99

I believe the 4,294,967,295 -gon is the largest odd sided n-gon that can be constructed. It a Fermat number (2^2^5 -1), and it is the product of the primes 641 and 6700417.

 

[lpetrich]

#99

Spoiler

Being constructible with ruler and compass translates into algebra as being constructible with a finite number of basic arithmetic operations and square roots starting with integers. Construction of a regular n-gon requires finding trigonometric-function values, and one may conveniently start with $\cos (\pi/n)$ and determine $\sin (\pi/n)$, $\cos (2\pi/n)$, and $\cos (2\pi/n)$ using basic arithmetic and square roots. One does so by solving $T_n(x) = -1$ where $x = \cos (\pi/n)$ and $T_n(x)$ is the Chebyshev polynomial of the first kind.

There is a theorem that this Chebyshev-polynomial equation can be solved with basic arithmetic and square roots only if the number of sides n has form $n = 2^m \cdot p_1 \cdots p_k$ where $p_1 \cdots p_k$ are distinct Fermat primes. They have the form $F_n = 2^{2^n} + 1$, and only five of them are known, for n = 0, 1, 2, 3, and 4: 3, 5, 17, 257, and 65537. All the others, up to n = 32, are known to be composite, and it is not known whether or not there are any more prime ones.

Turning to the problem statement, n = 4,294,967,295, I find that it is 3 * 5 * 17 * 257 * 65,537. I used Mathematica's FactorInteger[] function to do the factoring. Since all five factors are distinct Fermat primes, it is thus possible to construct a polygon with that many sides with ruler and compass.

 

[lpetrich]

#100

Spoiler

Make it in the shape of a plus sign:

 # 
###
 #

 

[bluej]

#99 : I would simply draw a circle.
$\frac {2\pi} {4294967295 }$ is an angle too small for the human eye to differentiate the polygon side from a smooth curve on a piece of paper.
Of course, using an abstract compass and straight edge and open intergalactic space - then the polygon side might not look like a smooth curve anymore, to the human eye, that is

 

[fresh_42]

101. At each station, tickets are sold to every other station. Then the route network will be expanded and 34 new tickets will be added. How many stations are there?

(99. was a joke.)

D108

 

[mfb]

We go from n(n-1) tickets to (n+k)(n+k-1)=n(n-1) + 2kn + k² - k tickets.
34 = 2kn + k² - k
Solve for n: n = (34+k-k²)/(2k) implies k must be a factor of 34=2*17 and smaller than 7, that means k=1 or k=2.
k=1 leads to n=17: We add 17 tickets to the new station and 17 tickets coming from the new station.
k=2 leads to n=8. We add 8*2 tickets going to the new stations, 8*2 tickets from the new stations to old stations, and 2 tickets between the new stations.

8 or 17 existing stations.

 

[lpetrich]

#101

Spoiler

Each kind of ticket has a departure station and an arrival station, and they are different from each other. That means that the total number of tickets is $N(n) = n(n-1)$. So we must find some old number of stations n₁ and some new number n₂ such that $N(n_2) - N(n_1) = 34$. This equation becomes $(n_2 - n_1)(n_1 + n_2 - 1) = 34$.

Since 34 is 1*34 = 2*17, I try each combination of factors. The two solutions are n₁ = 17, n₂ = 18 and n₁ = 8, n₂ = 10.

 

[fresh_42]

102. A horse, a giraffe and an elephant have agreed to make a race. It's over $1,000$ meters, and only two animals compete against each other per run.

In the first race, the horse defeats the giraffe. The moment it crosses the finish line, the horse is $100$ meters ahead. Race two is won by the giraffe - $200$ meters ahead of the elephant.

Finally, the horse and the elephant compete against each other. How far will the horse be ahead when it crosses the finish line?

D108

 

[KnotTheorist]

Spoiler: 102 Spoiler

Turning the problem into equations, we have
(1) 1000 = R_h * t_h
(2) 900 = R_g *t_h
(3) 1000 = R_g *t_g
(4) 800 = R_e * t_g
and we need to solve for x in these two equations
(5) 1000 = R_h * t_h
(6) 1000 - x = R_e * t_h

Dividing the 1st equation by the second and third by the forth, we get,
10/9 = R_h/R_g and 10/8 = R_g/R_e
Multiplying these together, we get 100/72 = R_h/R_e
Then dividing the the fifth equation by the sixth, we get,
1000/(1000-x) = R_h/R_e = 100/72 (from our previous ratio)
Thus 1000/(1000-x) = 1000/720
This x = 280 meters

 

[fresh_42]

103. Arrange $1,1,2,2,3,3,4,4$ in a way that between the ones is exactly one of the other numbers, between the twos two other etc.

D108

 

[KnotTheorist]

Spoiler: 103 spoiler

41312432

 

[fresh_42]

104. How many different (connected) plane figures can be created with five squares?

D108

 

[fresh_42]

105. Why equal all sums of any diagonal which same number?
$$
\begin{bmatrix}
9&14&21&35&11&7\\
13&18&25&39&15&11\\
21&26&33&47&23&19\\
39&44&51&65&41&37\\
33&38&45&59&35&31\\
25&30&37&51&27&23
\end{bmatrix}
$$
(A diagonal of a matrix is a set where one matrix entry from each row and each column is chosen: $a_{j \pi(j)}$.)

D109

 

[mfb]

Spoiler: 105

$a_{ij}=b_i+c_j$ for tuples b,c. Any diagonal sums to $\sum b_i + \sum c_j$.

 

[fresh_42]

106. Someone has walked south for five kilometers, then five kilometers to the west, and finally five kilometers to the north, to return to their starting point. All the same, he was not at the North Pole.

Where else is this possible?

D109

 

[SSequence]

106. Someone has walked south for five kilometers, then five kilometers to the west, and finally five kilometers to the north, to return to their starting point. All the same, he was not at the North Pole.

Where else is this possible?

South Pole? Like it seems that the second movement (the one in west) is sort of redundant (in the sense that it can be removed or the distance walked can be changed)? Or am I missing something too obvious?

 

[KnotTheorist]

Spoiler: 106 spoiler

He would be somewhere close to the south pole. There are actually a family of points that satisfy these conditions.
You start somewhere north of the south pole, travel south 5km, then walk around in a circle 5km such that you have moved an integer number of circumferences about your current great circle (you are back in the same spot after moving South) then you travel 5km North.

 

[SSequence]

Nice point. Shouldn't there also be a possibility of other solutions if we are allowed to adjust the radius R (or the distance walked for that matter)?

 

[fresh_42]

107. A quadrilateral piece of paper is cut into six pieces with two straight cuts. The paper is neither bent nor is it folded. In addition, the pieces of paper must not be rearranged or superimposed after the first cut.

How can that be?

D109

 

[jbriggs444]

107. A quadrilateral piece of paper is cut into six pieces with two straight cuts. The paper is neither bent nor is it folded. In addition, the pieces of paper must not be rearranged or superimposed after the first cut.

How can that be?

D109

Spoiler

quadrilateral ABCD where AB and CD intersect in their interiors?

 

[fresh_42]

Spoiler

quadrilateral ABCD where AB and CD intersect in their interiors?

A double triangle merged in one corner?

 

[jbriggs444]

A double triangle merged in one corner?

Yes. But I see that it can also be achieved with a V-shaped quadrilateral.

Spoiler

 

[fresh_42]

There is also a solution with a proper quadrilateral.

 

[KnotTheorist]

Spoiler: 107 spoiler

[/SPOILER

 

[fresh_42]

108. We are looking for the smallest eight digit natural number, where all digits are different, and the number is divisible by $36$.

D109

 

[SSequence]

OK I think my answer in post#440 is supposed to be wrong (because I was confusing how the directions are supposed to work). But there is still something that is bit ambiguous for me.

Suppose you were at a distance of 5km from southpole. You move 5km to south and reach the southpole. Now is the movement of "5km to the west" supposed to keep you at the same point? I would guess, so in a way since for all other points (except northpole) the movement towards west is making one move in a circle?

If the convention in previous paragraph is supposed to hold, then the distance of 5km from southpole should also be a solution. Right?

============================

Also, another small question (for my own understanding regarding directions). If I am at southpole then what does movement such as "5km towards south" would mean?