Solution To Mensa 3D Geometry Circle Problem

 

Let O be the center of the cube after envisioning the two spheres centered at O (passing through the 2 circles) and sketched in two dimensions (as shown above).  Then the sphere that passes through the inscribed circle has radius:

s Ö (2)/ 2 

The larger sphere that passes through the inscribed circle has radius:

 s Ö (3)/ 2 

The  distance between the 2 circles cannot be smaller than the distance between the spheres. Thus, for any point P on the circumscribed circle and any point Q on the inscribed circle:  

PQ  >  (s/ 2) (Ö (3)  -  Ö (2) )

To show that this lower bound is attained, envisage a ray from O aimed toward the point where the inscribed circle is tangent to the face on which the other circle is circumscribed (as depicted in the revised 2D sketch).  Initially this ray passes through the interior of the circumscribed circle.

Now, visualize rotating the ray while keeping it tangent to the inscribed circle. Eventually, the ray will rotate far enough that it no longer passes through the interior of the circumscribed circle. By the principle of continuity, it follows that in some intermediate position the ray touches both circles.

Let P be the point where the ray touches the circumscribed circle and Q be the point where the ray touches the inscribed circle.  The points O, P and Q are co-linear and:

PQ = (OP - OQ) =  sÖ (3)/ 2)  -  sÖ (2)/ 2) 

= (s/ 2) {Ö (3)  -  Ö (2)}  

Which is the minimal distance between the two circles.

Reviewing Algebra II - Part Two: Exponents & Solving Exponential Equations

  Simplifying and solving exponential equations is one of the more important aspects given the wide array of applications in the physical sciences, such as radioactive decay, obtaining the half life of an isotope etc. So in this Part Two we focus on reviving the skills involved in solving these equations - which are  fundamental part of Algebra II- or learning them for the first time. Again, this is also based on the rules and properties of manipulating exponents which are generally part of Algebra I.   For example, we already saw in the previous post one of these rules applied to quotients, i.e.

a ^m /a ⁿ     =  a ^{m - n}

^{So for:} 

a ⁵ /a ³   we get:    a ^{5 - 3}    =  a 

Other rules for exponents:

(i)  a ^-n  =  1/a ⁿ

(ii)   (a ^m) ⁿ     =  a ^m n 

(iii)   a ^1/n    =   ⁿ Ö a

(iv)   a ^m/ n     =   (a ^1/n ) ^m     = (a ^m) ^1/n

=  ( ⁿ Ö a ) ^m     =   ⁿ Ö( a ^m )

There are generally 2 ways to solve exponential equations: i) where both sides of the equation have the same base, and (ii) when they have different bases.

For case (i) consider:   

3 ^x =  3 ^{2x - 2}

^{Here the same base (3) allows us to solve for x by equating the exponents:}

^{x =  2x - 2} 

^{so:  x = 2   which can be checked by plugging x = 2 back into the eqn.  Then:}

^{3 2 =  3 2(2) - 2}

3 ² = 3²

When different bases are involved, we have basically 2 choices: a) work to get both sides to the same base, then proceed as shown for case (i) or b) crank out the needed value using logs, say base 10 or natural. Since most scientific work is now with the latter we use that:  For example, take the equation:  4^x = 15

We begin by taking the natural logarithm of both sides:

ln 4^x   =   ln 15   

Then use the power rule: i.e. ln b^x   = x ln b, to bring the variable exponent (x) to the front, i.e.

x ln 4 =  ln 15

We then can solve by dividing both sides by ln 4:

x =    ln 15/  ln 4  =   2.708/ 1.386  = 1.953

The other approach, usually easier, is to bring each side of an exponential equation to the same base. For example, say: 5^{x -1} = 0.04

The trick here is to get both sides in base 5 and go from there. It helps if we can see the decimal 0.04 can be expressed as a fraction, 0.04 = 4/ 100.

But 4/100 is just 1/ 25 and we immediately see  5² =  25, and we have our base 5.

Then,  working through we get:  5^{x -1} = 1/ 4^x = 15
 =    5^-2

So we can now equate the exponents:  x - 1 =  -2 or x = -2 + 1  = -1

Practice Problems: 

Solve each equation for the unknown and check:

 (1) 2a^-3 - 1 = 15

(2) 9^x =  55

3) 5x^1/2 + 7   = 22

4) 3a³   = 81

5) 4^x = 8 ^x+1

6) b^-5 = 1/32

 

7)  3 +   7 ^x-1  =  10

For each of the following, write as a power with positive exponents in simplest form:

8)

9)

10)

Holman Jenkins Jr's Solution For 'Young Socialists' Seeking Relief: "Join The Army!"

 

 

Holman Jenkins Jr. -he whose bellowing twaddle dwarfs his actual mental capacity- again can't help himself WSJ, ('The Socialism We Already Have', Aug. 26, p. A13).  Therein, he not only conflates socialism with 'property tax breaks' (for homeowners over 65) but also "redistributing resources from young to old"- a tired trope I already demolished in two previous posts, e.g.

NY Times' Douthat (& Yale's Moyn) Wrong As Ever About "Older Americans Hoarding The Future"

And:

Yale Econ Prof Claims "Older People" in U.S. Are A "Gerontocracy Stealing From the Young" - Where and How He's Off Base

But what would you expect from an op-ed writer so Trumpian he lies like he breathes, and gets away with a lot because he's confident few will check his howlers and detritus. Like when he claimed in a Mar. 22-23 WSJ column ( 'CBS Kicks The UFO Habit', p. A13)  the Nimitz F-18 pilots were actually chasing "a 46' Chinese drone", not a genuine UFO -reported by Ret. Lt. Cmdr. David Frav0r as an incredibly high velocity craft reaching the cap point in seconds-  e.g. his House testimony.   

Or, when he denied in one nutso column from 2020 ('Disinfecting Journalistic Ethics',. WSJ, May 6, p. A13)  that Dotard ever stated people ought to consider 'disinfectant'.   Not prepared for the actual video that circulated contradicting his lie, i.e.

https://www.youtube.com/watch?v=M9kWsUwTcj8

But look, this is how Holman rolls, and he hasn't let up since actively pushing his 'Russia Hoax' baloney for the last x years. Despite the fact that former AF officer Reality Winner ended up in a federal slammer for 5 years  (under the Espionage Act) for leaking docs pertaining to  Russian interference in the 2016 United States elections.   Which, of course, Jenkins has termed "a hoax". 

In his latest  column, Jenkins continues his deformations of the truth often mixing them with half-truths to try to gain more traction among the weaker minded. He is totally convinced that if he offers enough 'loaded' malarkey combined with codswallop he can get enough WSJ readers to bite.  For example, quoting Richard Posner's book, 'Aging and Old Age' from which he unloads this chestnut: "Posner gives a frank discussion of the relationship between the short time horizon in the elderly and unchecked selfishness."

So Jenkins Jr. like Parsons commits the same logical fraud: conflating two groups of elders - one in the elite or upper 1 percent (like Steve Forbes and Warren Buffet).  The others in the lower 30 or 40 percent just scraping by with Social Security and some benefit from reduced property taxes- because of the Homestead Exemption in their states. (Btw, the reduced property taxes for those over 65 do have a rationale - because they no longer have children who need to attend schools in their districts - which property taxes pay for).

But the biggest outrage is when Holman stretches his definition of socialism to give a twisted bit of advice to the "young socialists seeking relief from having to be self supporting".  Offering this:

"For them, there's always the Army.  They'll know when their next meal is coming from and what to wear. And accommodations will be provided."

Adding: "Of course, the U..S. military prizes initiative. A lesson for the young socialists: You can't have everything."

Does this fulsome twit merit a rap on his shroom-gobbling noggin or what? 

See Also:

Social Security

Extremist Billionaire Media Owners Want to Steal Your Social Security So They Don’t Have to Pay Slightly Higher Taxes

by Dean Baker | August 28, 2026 - 5:01am | permalink

— from Beat the Press

An extreme position does not become less extreme just because someone can put forward one that’s even more extreme. Massacring 100 children doesn’t become a moderate position just because someone is advocating killing 200 children.

This is how we should view the line being pushed by “moderate” voices that we have to deal with the $40 trillion debt with both spending cuts and tax hikes. The reality is that, apart from the military and Homeland Security, there is little fat in spending to be cut, as even Elon Musk inadvertently acknowledged. Insofar as we have a deficit problem, the issue is on the tax side, as can be easily shown. The rich have been taking an ever-larger share of national income over the last half-century, and they don’t feel like paying taxes on their winnings.

» article continues...

And:

Wealth Inequality

Is It Socialism Or Just Common Sense?

by Chuck Collins | August 27, 2026 - 4:57am | permalink

— from OtherWords

The debate over capitalism vs. socialism often looks like a battle of cartoon caricatures: Monopoly Man versus the Marx Monster.

Across the planet there are countless flavors of capitalism and socialism, unique mashups of free markets and government policies. Into this muddle wade the right-wing ideologues who fear the red “socialist” demon hiding in every corner.

If Fox News existed in the 1950s, they would’ve labeled Republican President Dwight Eisenhower a socialist for taxing the wealthy and making broad public investments to expand homeownership, enable debt-free higher education, and build infrastructure and technology that expanded the modern middle class.

Yet, what do these labelers call modern-day China, a country that mints hundreds of billionaires a year and dominates global markets with private production? Are they socialist or capitalist?

» article continues...

And:

Corporate America | Economic Policy

For May Day: The System We Need

by Robert Reich | May 1, 2026 - 5:03am | permalink

— from Robert Reich's Substack

David Zaslav, CEO of Warner Bros. Discovery, has reason to smile

Friends,

Warner Bros. Discovery shareholders voted last Thursday on the Ellison family’s purchase of the company. Some 1.743 billion shares were cast in favor of the sale; 16.3 million were cast against it, a ratio of roughly 99 to 1.

1. Great for a Handful of Super-Wealthy, but Bad for Workers and Bad for America

This vote came soon after more than 4,000 workers in the media industry — directors, screenwriters, producers, actors, editors, cinematographers, musicians, and composers — signed a letter predicting an industry disaster if the sale went through.

» article continues...

And:

Youtube videos:

Samuel Moyn: How did we end up with political gerontocracy

&

Anna Gát and Samuel Moyn: Knowing When to Leave? The Power of Old Men - YouTube

NASA's Just Launched Nancy Grace Roman Telescope Set To Begin Best Astronomy Era Yet

 

    SpaceX Falcon Heavy rocket launches NASA's Nancy Grace Roman Telescope  Sunday

The National Aeronautics and Space Administration launched a new flagship telescope yesterday: a powerful observatory that will study dark matter and hunt for exoplanets, from an orbit about one million miles from Earth.

A SpaceX rocket took off with the $4.6b instrument, called the Nancy Grace Roman Space Telescope, just before 7:30 a.m. ET Sunday from a launchpad in Florida. The launch went smoothly, setting up the 8m aperture telescope to fly out to the L2 Lagrange point. This is one of 5 Lagrange points, named after the mathematician Joseph-Louis Lagrange, e.g.

As seen in the graphic above, three Lagrange points (L1, L2 and L3) lie along the line of centers connecting the two major bodies, in this case the Sun (bright center) and Earth.  These are called "unstable" Lagrange points, while L4 and L5 are stable points.  All the points represent locations in space where the gravitational forces of a 2-body system (e.g. Sun-Earth) produce enhanced regions of attraction and repulsion. Locating a point of 'balance' means a spacecraft can then remain in position and reduce fuel expenditure. 

As NASA reports:

Roman pairs a large field of view with crisp infrared vision to explore vast swaths of the sky and probe deeply into cosmic history. This flagship mission will help astronomers explore dark matter, dark energy, and worlds outside of our solar system, known as exoplanets. Its surveys will support a broad range of research extending far beyond the mission’s main science goals.

“Roman is exactly the kind of success story we want to see across NASA,” said NASA Administrator Jared Isaacman. Adding:

“Roman will give us a new atlas of the universe, push the boundaries of discovery, and demonstrate what is possible when America’s space program pairs bold ambition with disciplined execution.”

In the words of Nicky Fox, associate administrator for the Science Mission Directorate at NASA Headquarters in Washington.

“With its large field of view and fast survey speeds, Roman will usher us into a new era of discovery and make the invisible visible, setting the foundation for humanity’s search for life beyond our solar system.”

During launch and early orbit, Roman uses the Near Space Network’s ground stations and relay satellites to exchange tracking, telemetry, and command data with ground controllers. About 70 minutes after launch, the Deep Space Network takes over communications and guides Roman toward the second Sun-Earth Lagrange point, or L2, about one million miles from Earth. 

Ironically, or perhaps not, Trump wanted to cancel this once- in -a lifetime project to "save money". (As if he knew anything about that!) Fortunately, according to a WSJ piece, more sober heads in Congress weren't having it and allowed it to go forward.  Amazing what some courage and a working spine can do!

  For sure the Grace Roman Telescope's orbital position at the L2 Lagrange point (which it will reach in about 100 days) will afford us access to previously undetected exoplanets, perhaps as many as 200,000 more - as well as possibly piercing the physical mystery underpinning dark matter and dark energy and their relationship.  I look forward - at least so long as I am around - to experiencing one of the most productive and exciting astronomy periods ever.

Stay tuned! 

See Also:

Fantastic New Images of Distant Cosmic Objects Prove Webb Telescope's $10b Worth

And:

The Vera Rubin Observatory Unveils New Astronomical Frontiers With A New Kind Of Telescope

And:

Alien Worlds: The Top Exoplanet Discoveries of 2013 | Space

And:

A Possible Exoplanet In Another Galaxy Identified Using Chandra X-Ray Telescope Records

And:

YouTube video:

Something Left the Sun. It May Reach Earth Tonight

Phase Spaces In Plasma Physics (Part 2): The Liouville Equation

 Recall in Part 1, we saw the density of systems for the (x₁, v₁) phase space is:

N(x₁ ,v₁ ,t) = d (x₁ -  X₁ (t)) d (v₁ -  V₁ (t))

Where:  x₁ = X₁ (t)),  v₁  = V₁ (t)) 

Analogously, in the 12-dimensional phase space (two particles considered) we saw from the Part 1 solutions (#2):  

(x₁, v₁, x₂, v₂ ) =  ( _x1 , y₁ , z₁, _vx1, _vy1,  _vz1, _x2 , y₂ , z₂, _vx2, _vy2,  _vz2,) 

And there is one system occupying the point:

x₁ = X₁ (t)),  v₁  = V₁ (t), x₂ = X₂ (t)),  v₂  = V₂ (t) i.e. at time t.  The density of systems in this phase space is then:

N( x₁, x₂ ,v₁ , v₂ ) =

d (x₁   -   X₁ (t)) d (v₁   -  V₁ (t))d (x₂   -    X₂ (t)) d (v₂   -   V₂ (t))  

Basically, then there is one system in 6No  -dimensional space, so by analogy with the density of systems expression e.g. for N( x₁, x₂ ,v₁ , v₂ ), we can write:

N(x₁, x₂ ,v₁ , v_{2 ....} x_No ,v_{No ,} t) = å^No _i=1      d (x_i   -   X_i (t)) d (v_i  -   V_i (t))

As with the Klimontovich equation, the Liouville equation is found by taking the time derivative of the appropriate density.  Given the already defined  density of systems is the product of 6No  terms then the time derivative must involve the product of  6No  terms.  We can use the expression:

¶ / ¶ t   d [ (x₁   -   X₁ (t)]  =  -   ¶   X_i/ ¶ t ·  Ñ x i  d [ (x₁   -   X₁ (t)] 

The time density will be:

¶ N / ¶ t +  å ^No _i=1  V_i (t)  ·  Ñ x i   å^No _j=1      d (x_j   -   X_j (t)) d (v_j  -   V_j (t))  +

å^No _i=1    V'_i å^No _j=1      d (x_j   -   X_j (t)) d (v_j  -   V_j (t))  = 0

The next standard step employed by plasma physicists is to use the identity:  ad (a  -  b)  =  bd (a  -  b) to replace V_i   by v_i

Leading to:

V'_i (t) = q _si / m _si  [ E (x_i,t) + V_i /c  x  B( x_{i ,}t)]

Given that the products are just the density of systems, N, the equation for the time derivative of N(x₁, x₂ ,v₁ , v_{2 ....} x_No ,v_{No ,} t) 

becomes:

¶ N / ¶ t +  å ^No _i=1  V_i · Ñ x N +å ^No _i=1  V'_i (t)  ·  Ñvi N  = 0

which is the Liouville equation, and when combined with Maxwell's equations, viz. 

i)  Ñ X H  =  J    + ¶D / ¶ t

ii)             Ñ X E  =  - ¶B / ¶ t

iii)           Ñ · B  =  0   

iv)       Ñ · D  =  r    

is an exact description of a plasma.