mathematics - The largest Thursday number

A Thursday number is a number $N$ where any three consecutive digits make a prime, and all such primes formed are distinct.

For example, $13739$ is a Thursday number because $137$, $373$ and $739$ are all distinct primes.

Find the largest Thursday number. (Preferably without using a program)

P.S. I know today is not yet Thursday, but I may not be able to come online tomorrow.

thermodynamics - Why must the particles of an ideal gas be point-like?

Why is a gas of elastically colliding hard balls of finite size not ideal?

Respectively:

Why is it essential that the particles of an ideal gas are point-like?

Especially:

Which differing thermodynamic properties of a gas of not point-like particles are most striking in comparison with an ideal gas (of point-like particles)?

game theory - Tic-tac-toe 1000 in a row

Consider a variant of Tic-Tac-Toe and Gomoku on a board that's infinite in every direction. X wins by getting a line of 1000 X's horizontally, vertically, or diagonally. O cannot win but plays for a draw by preventing X from winning indefinitely. X gets to move twice each time and starts, so the turn order goes XXOXXOXXO....

Can X win with perfect play?

Answer

Yes, X can win.

To simplify things I'm going to take advantage of your rules that O cannot win, so that X doesn't need to worry about O getting 1000 in a row.

Just consider a 1 dimensional game, chose an origin, and label the locations with integers in order.

Define the "bin $K$" as the set of spaces $x$ such that $1000\ K \le x < 1000 (K+1)$.

A simple winning strategy is:

• 
  

  For the first $M$ turns, have X play each piece in a previously empty bin.

  
  


• 
  

  Now while possible, have X play each piece in a bin with only one of its pieces and no O pieces.

  
  


• 
  
  

  Now while possible, have X play each piece in a bin with only two of its pieces an no O pieces.

  
  


• 
  

  And so on.

  
  

It should be intuitively clear that X will always win, with $M$ chosen sufficiently large.

To prove this without relying on intuition, I only need to show there is an answer to

What value of $M$ is sufficient if the win condition for X is to get $Z$ pieces in-a-row?

After $M$ turns, X will have a piece in $2M$ bins, and O can block at most $M$ of them.

After the next $M/2$ turns, X will have two pieces in $M$ bins, and O can block at most $M/2$ of them.

More generally, after $\sum_{t=0}^N M/2^t$ turns, X will have $(N+1)$ pieces in $M/2^N$ bins. Note that $M$ must be at least $2^N$ for the last number of turns in that sequence to be an integer. So $M \ge 2^N$.

So a sufficient value is: $$Z = N+1 \quad\quad \rightarrow \quad\quad M \ge 2^{Z-1} $$

If the rules are changed such that O can win with 1000 in a row, then the math is a bit more complicated. But you could imagine X infrequently using a turn here or there to stop O and then continue on with the plan. X should still be able to win.

general relativity - Can the Hubble constant be measured locally?

The Hubble constant, which roughly gauges the extent to which space is being stretched, can be determined from astronomical measurements of galactic velocities (via redshifts) and positions (via standard candles) relative to us. Recently a value of 67.80 ± 0.77 (km/s)/Mpc was published. On the scale of 1 A.U. the value is small, but not infinitesimal by any means (I did the calculation a few months ago, and I think it came out to about 10 meters / year / A.U.). So, can you conceive of a measurement of the Hubble constant that does not rely on any extra-galactic observations?

I ask because, whatever the nature of the expansion described by the Hubble constant, it seems to be completely absent from sub-galactic scales. It is as though the energy of gravitational binding (planets), or for that matter electromagnetic binding (atoms) makes matter completely immune from the expansion of space. The basis for this claim is that if space were also pulling atoms apart, I would naively assume we should be able to measure this effect through modern spectroscopy. Given that we are told the majority of the universe is dark energy, responsible for accelerating the expansion, I wonder, how does this expansion manifest itself locally?

Any thoughts would be appreciated.

Answer

Everything doesn't expand equally because of cosmological expansion. If everything expanded by the same percentage per year, then all our rulers and other distance-measuring devices would expand, and we wouldn't be able to detect any expansion at all. Actually, general relativity predicts that cosmological expansion has very little effect on objects that are small and strongly bound. Expansion is too weak an effect to detect at any scale below that of distant galaxies.

Cooperstock et al. have estimated the effect for systems of interest such as the solar system. For example, the predicted general-relativistic effect on the radius of the earth's orbit since the time of the dinosaurs is calculated to be about as big as the diameter of an atomic nucleus; if the earth's orbit had expanded according to the cosmological scaling function $a(t)$, the effect would have been millions of kilometers.

To see why the solar-system effect is so small, let's consider how it can depend on $a(t)$. There is a cosmology called the Milne universe, which is just flat, empty spacetime described in silly coordinates; $a(t)$ is chosen to grow at a steady rate, but this has no physical significance, since there is no matter that has to expand like this. The Milne universe has $\dot{a}\ne 0$, i.e., a nonvanishing value of the Hubble constant $H_o$. This shows that we should not expect any expansion of the solar system due to $\dot{a}\ne 0$. The lowest-order effect requires $\ddot{a}\ne 0$.

For two test particles released at a distance $\mathbf{r}$ from one another in an FRW spacetime, their relative acceleration is given by $(\ddot{a}/a)\mathbf{r}$. The factor $\ddot{a}/a$ is on the order of the inverse square of the age of the universe, i.e., $H_o^2\sim 10^{-35}$ s$^{-2}$. The smallness of this number implies that the relative acceleration is very small. Within the solar system, for example, such an effect is swamped by the much larger accelerations due to Newtonian gravitational interactions.

It is also not necessarily true that the existence of an anomalous acceleration leads to the expansion of circular orbits over time. An anomalous acceleration $(\ddot{a}/a)\mathbf{r}$ just acts like a slight repulsive force, which is equivalent to reducing the strength of the gravitational attraction by some small amount. The actual trend in the radius of the orbit over time, called the secular trend, is proportional to $(d/dt)(\ddot{a}/a)$, and this vanishes, for example, in a cosmology dominated by dark energy, where $\ddot{a}/a$ is constant. Thus the nonzero (but undetectably small) effect estimated by Cooperstock et al. for the solar system is a measure of the extent to which the universe is not yet dominated by dark energy.

The sign of the effect can be found from the Friedmann equations. Assume that dark energy is describable by a cosmological constant $\Lambda$, and that the pressure is negligible compared to $\Lambda$ and to the mass-energy density $\rho$. Then differentiation of the Friedmann acceleration equation gives $(d/dt)(\ddot{a}/a)\propto\dot{\rho}$, with a negative constant of proportionality. Since $\rho$ is currently decreasing, the secular trend is currently an increase in the size of gravitationally bound systems. For a circular orbit of radius $r$, a straightforward calculation (see my presentation here, sec. 8.2) shows that the secular trend is $\dot{r}/r=\omega^{-2}(d/dt)(\ddot{a}/a)$. This produces the undetectably small effect on the solar system referred to above.

In "Big Rip" cosmologies, $\ddot{a}/a$ blows up to infinity at some finite time, so cosmological expansion tears apart all matter at progressively smaller and smaller scales.

Cooperstock, Faraoni, and Vollick, "The influence of the cosmological expansion on local systems," http://arxiv.org/abs/astro-ph/9803097v1

Copycat chess is back

This puzzle is inspired by and has similar rules to the series of puzzles by Sleafar.

Definition. A copycat chess opening is a sequence of moves starting from the conventional starting position, where every move by White (including the last one) is copied by Black identically, resulting in a symmetrical position with respect to the mid-horizontal axis. Every move must be legal, but the last move doesn't have to be checkmate, hence the name opening instead of game.

What is the shortest copycat chess opening in which a rook captures another rook?

Answer

How about seven moves each

1) p-a4 p-a5 2) p-b4 p-b5 3) p(b)xp pxp 4) p-a6 p-a3 5) p-a7 p-a2 6) pxN=R pxN=R 7) RxR RxR

astronomy - How do you measure distance to stars within the galaxy?

I know that for close by stars (<50 LY) we can use the parallax effect. And for distant galaxies we use red-shift (& hubble's constant). So how do we measure how far is a star lets say 50,000 LY from the earth?

I know I am missing something, I just don't know what.

How can we assume there is a relationship between red-shift and distance when the stars

• act like a (turbulent) fluid within the galaxy and


• can be moving in any sort of different manner?

Edit: It is a physics question, because I really want to know what is the star velocity model within the galaxy in order to use red-shift. My instinct tell me that stars move like a swarm within the galaxy with an overall rotation about the galaxy center of gravity (In agreement to the edited horowitz answer).

Answer

There are numerous distance indicators used for within the galaxy. The most common way is by using intrinsic magnitude. By knowing how bright an object would be if we were close, we can determine how far away it is by how dim it is. There are many types of stars where we have a rough idea of how bright they should be due to characteristics of the star:

1. 
   

   Cephied Variables: The original type of variable star that was used by Hubble to determine the distance to the Andromeda Galaxy.

   
   


2. 
   

   RR Lyrae Variable: Like the Cephied variable, but usually dimmer.

   
   
   


3. 
   

   Type 1a Supernova: These guys, unlike the first two, are cataclismic variables. Essentially a binary white dwarf slowly accretes matter from its binary till it reaches the Chandrashankar Limit, after which point it explodes in a very characteristic way (since the mass at the time of explosion is roughly constant).

   
   


4. 
   

   Main Sequence Stars: Generally less accurate than the first 3, there are some types of main sequence stars which are used to find distances in a similar way.

   
   

There are a few other ways we can measure distances:

Perpendicular Movement: For example there is a "light echo" from SN 1987A which is essentially light from the supernova interacting with dust around the old star. Since this echo should be expanding at the speed of light, we can tell how far away the nova is by the angular velocity of the light.

Relative Velocity in a Moving Cluster: (see dmckee's answer)

Tulley-Fisher relation: A relationship between the luminosity of the galaxy and it's apparent width. Can be used as a decent distance calculator.

Faber-Jackson Relation: Similar to Tulley-Fisher, relates luminosity with radial velocity dispersion rate.

EDIT: Some more information about redshifts.

The whole relationship between redshift and distance was in fact established by Hubble by relating distance to Cephied variables (I believe) with redshift. Later on it was made more precise using supernova, which are brighter and can be seen from much father away (I think recent supernova can be occasionally seen around Z=2, while Cephieds are all Z<1). Within a galaxy, redshift cannot be used directs since the "peculiar velocity," the velocity within the galaxy, completely overshadows the effects of universe expansion on which Hubble's Law is based. Redshift within the galaxy is useful for certain other techniques.

EDIT: corrected a few minor errors.

cipher - Room 5 of the Maze

This is Room 5 of the Maze series. For those wishing to start at the beginning, click here. To go back to the previous room, click here.

As you arrive at the answer to Room 4, you realize it's the same one that you got for Room 3. With a sudden flash of insight, you go back to the keypad and scrutinize it. The faint fingerprints you see are on the keys 0,1,3 and 7. You realize with dread that you have been in this room before (0 from your unfortunate predecessor, 33 from your peer Anne whose skin is still recovering from the Itchyverse, and the rest from your previous entry. But wait! You look to the wall next to the keypad for the scratch marks that were there before and you see none. Were they ever really there? You look around, bewildered, for an explanation and you see nothing but the cold walls and the suspended platform.

Eventually you hopelessly resign yourself to foraging ahead and enter your answer into the keypad. As usual, the door disappears and you find yourself in Room 5. You don't even think about resting--what would be the point in trying, you know you can longer get any sleep in this Maze. You just walk helplessly towards the platform and grab the note. You stop before reading it and in defiant anger you rip a piece off and throw it to the ground. Before it can even hit the ground, it evaporates into thin air. You realize what you've just done and force yourself to calm down before you ruin your hopes of escaping this place. You read the note:

Welcome to Room 5! If we must say, this is our personal favorite room; it was the first one that we created. As a result, it may turn out that it is actually easier than the previous--only time will tell. Good Luck!!

Key 1: >c 8DC

There is a rip in the paper after 'd' thanks to your moment of indignance. "I hope there was nothing too important on that bit", you mumble.

Key 2: ERLO GWTD ERW BLSH OFVVWWUWU ERW OLXERCSDH YRS RLAOWBN RTU OFMMBTHEWU ERW RWTQWHBG

Bad, bad test subject! We sure hope that you can complete this room without that missing information.

Did they know that you were going to rip the paper beforehand? Or did they make that information appear without you noticing? Your head begins to hurt from the constant confusion and you can feel your stability slipping. You need to hurry and escape this Maze!!!

A thorough investigation of the keypad shows only digits (with no discernible marks on any of them).

Answer from Previous Room:

117

EDIT: After a day of grueling attempts to find any significant meaning in the keys, a note appeared on the platform:

It seems that we had not properly balanced this room. That being the case, we will provide two pieces of information.
1. Key 1 has been put through one of your species' first recorded ciphers, though with an alphabet larger than the one originally used.
2. A fuller version of Key 2 is as follows: ERLO GWTD ERW NLDOE BLSH OFVVWWUWU ERW ERLDU OLXERCSDH YRS RLAOWBN RTU OFMMBTHEWU ERW NLDOE RWTQWHBG

This puzzle has now been solved. Here is the next, and final, room

Answer

It seems I did a pretty bad job on this room. After some consideration, I've decided to post the complete solution to this puzzle; while I indicated previously to f" that I would accept his answer if this went unsolved (and kudos to him for getting the correct final answer), I think it would be unwise to deprive current and future solvers the satisfaction of knowing how this puzzle was intended to be solved. Some may disapprove of this approach, but I think I'd rather leave the problem as is and post the information in this post rather than update it with, basically, instructions on how to solve this puzzle. All of that said, here goes.

First notice that Key 1 is a shifted Caesar cipher with the alphabet being ASCII instead of the usual one and the shift amount being ROT117, the answer from the previous room. This yields:

In CONGRESS, July 4th, 1776. We hold ... future security. Frequency, then o

This points to a passage from the preamble of the U.S. Declaration of Independence. Using this paragraph and analyzing the frequency of each letter (every one of the 26 occurs at least once in this passage), we get the following, sorted in descending order with ties broken by "o"rder of occcurrence (hence the lingering o in Key 1 before the rip; the whole word was order).

e (134), t (109), n (71), s (70), i (70), r (65), a (65), o (64), h (59), d (38), l (35), u (34), c (26), g (26), f (25), m (24), p (18), b (16), v (15), w (14), y (13), j (2), k (2), q (1), z (1), x (1)

Lining these up with the letters in order of occurrence in the passage (also what was meant by 'frequency, then order'), you get the substitution cipher of:

e, t, n, s, i, r, a, o, h, d, l, u, c, g, f, m, p, b, v, w, y, j, k, q, z, x
w, e, h, o, l, d, t, s, r, u, b, f, v, i, n, a, m, c, q, y, g, p, j, z, k, x

This is used to decipher Key 2 to the following:

THIS YEAR THE LION SUCCEEDED THE SIXTHBORN WHO HIMSELF HAD SUPPLANTED THE HEAVENLY

Which, as f" noted, is a reference to when Pope Leo I (the lion) succeeded Pope Sixtus III (the sixthborn, though the actual etymology of this name is believed to be the greek word for 'polished') who was preceded by Pope Celestine I (the heavenly). This took place in the year:

440

Giving us our final answer to enter into the keypad.