small brain teaser

Yesterday, I turned 46 years old. Next July, my wife, Barbara, turns 64. (No, you fans of Ben Franklin, they are not "so grateful").

Barbara noticed that, for a couple of months, the digits of our ages will be transposed. She said "that won't happen again."

Well, I pointed out: 75 - 57 = 18, 86 - 68 = 18, and being very optimistic, 97 - 69 = 18.
So the question:

1. (blindingly easy): what is the pattern?
2. (more challenging): what is going on here? Why does this work?

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Reference to Ben Franklin for those who don't want to follow the source:

Franklin had a friend who did not want to get married, but was battling with urges and lustful inclinations for the opposite sex. In the letter which follows, Franklin first advises the friend that the best solution for his urges is marriage. However, since he knows the friend will not take that advice, Franklin goes on to suggest that his friend have sexual affairs with old women. We know from the context that he is suggesting women over 45 years of age (see #3). His words about putting a basket over her head or turning out the light (see #5) illustrate an aspect of Franklin's character which is seldom exposed.

June 25, 1745
My dear Friend,
I know of no Medicine fit to diminish the violent natural Inclinations you mention; and if I did, I think I should not communicate it to you. Marriage is the proper Remedy. It is the most natural State of Man, and therefore the State in which you are most likely to find solid Happiness. You Reasons against entering into it at present, appear to me not well-founded. The circumstantial Advantages you have in View by postponing it, are not only uncertain, by they are small in comparison with that of the Thing itself, the being married and settled. It is the Man and Woman united that make the compleat human Being. Separate, she wants his Force of Body and Strength of Reason; he, her Softness, Sensibility and acute Discernment. Together they are more likely to succeed in the World. A single Man has not nearly the Value he would have in that State of Union. He is an incomplete Animal. He resembles the odd half of a Pair of Scissars. If you get a prudent healthy Wife, your Industry in your Profession, with her good Oeconomy, will be a Fortune sufficient.
But if you will not take this Counsel, and persist in thinking a Commerce with the Sex inevitable, then I repeat my former Advice, that in all your Amours you should prefer old Women to young ones. You call this a Paradox, and demand my Reasons. They are these:
1. Because as they have more Knowledge of the World and their Minds are better stor'd with Observations, their Conversation is more improving and more lastingly agreable.
2. Because when Women cease to be handsome, the study to be good. To maintain their Influence over Men, they supply the Dimunition of Beauty by the Augmentation of Utility. They learn to do 1000 Services small and great, and are the most tender and useful of all Friends when you are sick. Thus they continue amiable. And hence there is hardly such thing to be found as an old Woman who is not a good Woman.
3. Because there is no hazard of Children, which irregularly produc'd may be attended with much Inconvenience.
4. Because thro' more Experience, they are more prudent and discreet in conducting and Intrigue to prevent Suspicion. The Commerce with them is therefore safer with regard to your Reputation. And with regard to theirs, if the Affair should happen to be known, considerate People might be rather inclin'd to excuse an old Woman who would kindly take care of a young Man, form his Manners by her good Counsels, and prevent his ruining his Health and Fortune among mercenary Prostitutes.
5. Because in every animal that walks upright, the Deficiency of the Fluids that fill the Muscles appears first in the highest Part: the Face first grows lank and wrinkled; then the Neck; then the Breast and Arms; the lower Parts continuing to the last as plump as ever: So that covering all above with a Basket, and regarding only what is below the Girdle, it is impossible of two Women to know an old from a young one. And as in the dark all Cats are grey, the Pleasure of corporal Enjoyment with an Old Woman is at least equal, and frequently superior, every Knack being by Practice capable of Improvement.
6. Because the Sin is less. The debauching of a Virgin may be her Ruin, and make her for Life unhappy.
7. Because the Compunction is less. The having made a young Girl miserable may give you frequent bitter Reflections; none of which can attend the making an old Woman happy.
8. They are so grateful!!
Thus much for my Paradox. But still I advise you to marry directly; being sincerely Your affectionate Friend,
Benjamin Franklin.

"Numbers" and "largest primes"

While visiting one of my favorite political websites (commongroundcommonsense.org) I noticed a post that mentioned the TV series "numbers". The episode in question dealt with codes; in particular one can create a difficult to crack code by using a "key" which consists of the product of two very large prime numbers. The idea is that this key would take years and years of computer time to factor.

So, knowing various large prime numbers has some value.

But, the poster on this site (a poet) said something to the effect of finding the "largest prime number". Well, one thing that is easy to prove is that there is no largest prime number!

Here goes the proof:
suppose there is a largest prime number. Let it be "q".

Now consider the number p = q! + 1 (where q! means 1*2*3...*(q-1)*q, e. g., 6! = 1*2*3*4*5*6 = 720)
Now p cannot be prime as p > q and q was the largest prime. So p has at least one prime factor, call it r. r<= q hence r divides q!. r divides p.
So r divides (p - q!) = 1 which is impossible since no prime number divides 1.
Therefore there can be no largest prime number.

Another way of putting this is that if you know that q is a prime number, you can always find a prime number that is larger than q by finding the prime factors of q! + 1.

Of course, it can be very difficult to perform this factorization, but that is another matter.

A good teaser

Courtsey of a common ground, common sense member:

http://www.commongroundcommonsense.org/forums/index.php?act=ST&f=16&t=35269

There are five houses in a row in different colors. In each house lives a person with a different nationality. The five owners drink a different drink, smoke a different brand of cigar and keep a different pet, one of which is a Walleye Pike.

The question is-- who owns the fish?

Hints:
1. The Brit lives in the red house.
2. The Swede keeps dogs as pets.
3. The Dane drinks tea.
4. The green house is on the left of the white house.
5. The green house owner drinks coffee.
6. The person who smokes Pall Malls keeps birds.
7. The owner of the yellow house smokes Dunhills.
8. The man living in the house right in the center drinks milk.
9. The man who smokes Blends lives next to the one who keeps cats.
10. The Norwegian lives in the first house.
11. The man who keeps horses lives next to the one who smokes Dunhills.
12. The owner who smokes Bluemasters drinks beer.
13. The German smokes Princes.
14. The Norwegian lives next to the blue house.
15. The man who smokes Blends has a neighbor who drinks water.

Solution: (don't cheat! Hey, you don't get stronger by watching someone else lift weights, do you?)

http://www.geocities.com/onanyes/teaser.htm