Remember when Gas was Just $0.30/gal?

(By T4c. Messerlin. (Army) - http://www.dodmedia.osd.mil/Assets/1999/DoD/HD-SN-99-02404.JPEG, Public Domain, https://commons.wikimedia.org/w/index.php?curid=52903)

A friend recently sent me a greeting card with an insert showing what happened the year I was born (1959). Along with who was President (Dwight Eisenhower), Best Picture (Ben-hur) and top song (Lonely Boy by Paul Anka), the insert include the price of common items.

These prices provide a good check for what type of inflation we have had for the last 61 years. Lately, we have enjoyed several years of relatively low inflation (<= 3% since 2008). Going back to the period of 1973 to 1981, the rate was much higher, sometimes reaching over 10%.

A future price of an item, P₂, based on an original price, P₁, after undergoing n periods of inflation at rate, i (expressed in decimal form):

This formula can be used to calculate the inflation rate for each of the items listed.

 

Item	1959 Price	2020 Price	Inflation
Min. Wage	$1.00/hr	$13.00/hr	4.3 %
Gas	$0.30/gal	$2.89	3.6 %
House	$18,400	$578,000	5.8 %
Milk	$1.01/gal	$ 4.00	2.3 %
Eggs	$0.88/doz	$ 3.00	2.0 %
US Postage Stamp	$.04	$.0.50	4.2 %
Dow Jones Indus Ave.	679	26080	6.2 %

The last item listed is the Dow Jones Industrial Average stock index. Its average growth over the past 61 years was 6.2%. That's higher than any of the items listed including the house (the 2020 price was California median price for 2020).  The prices you currently pay are likely different based on your favored brands and location. So if you would like to check the rate for your items, please see the link where these calculations will be done for you. You just need to enter the two years of interest and the old and new prices. The spreadsheet will do the rest. 

Say Hello to Your Cousin

During the sheltering-in-place phase of the Covid-19 crisis, I've been reaching out to relatives across the country. We all have the same concerns of protecting family members who may be at risk or helping others who are struggling during the crisis. In the conversations, I discovered that I have a number of cousins working on family genealogy projects. It is interesting as one learns about their extended family trees. 

If two people at random meet, what are the chances the two are cousins? Very rare if you limit the term cousin to mean the two subjects have a common grandparent or great-grandparent. If we only go back two generations, our two subjects each have four grandparents and they are not related (as first cousins) if they have no common grandparent. We can continue this exercise, comparing great-grandparents, great-great-grandparents and so on. With each generation we go back, there is a doubling of potential ancestors of whom our two subjects may have in common. Eventually, every random pair of people will have one or more common ancestors thus making them cousins (although very distant). 

How many generations do we have to go back to guarantee this matching will occur? Making an assumption of one generation for every 25 years, and going back about 28 generations, brings us to the 14th century. Two raised to the power of 28 is 268 million. If our two subjects each have 268 million super-great-grandparents, that would require there be no overlap in any of the 526 million ancestors for the two subjects not to be related. The reason I took this exercise back 700 years, is the world's population at that time was only 300-400 million people which means the two people must be distant cousins.

Some have taken this topic much further using mitochondrial DNA analysis to estimate that determine all humans have a single common female ancestor, termed Mitochondrial Eve. 

(Image: screenshot from https://worldpopulationhistory.org/map/1302/mercator/1/0/25/)

Update (1/19/2021) - Numberphile has a video that explores the math of ancestors with similar results: Numberphile Ancestors.

Update (10/8/2022) - If you have blue eyes and meet someone else with blue eyes, you can also greet that person as your cousin. See this article reporting that everyone with blue eyes is a descendant of a single human from about 6000 - 10000 years ago.

One Hundred Trillion

My last post wrote about the approximate one trillion platelets in my blood donation. Another post described the power of compound interest in which a very modest investment could grow substantially over four hundred years at just a modest interest rate.

There is another example of when interest rates are very high. The enormous multiplying effect doesn't take hundreds of years - only months. That it what led the country of Zimbabwe to issue a one hundred trillion dollar note. At the peak of this period of hyper-inflation around 2007-2009, the cash in your pocket could be worth half, or less, than what it was the day before. In a single year, prior to the issuance of the note shown above, the country issued notes ranging from 10 dollars to 100 billion dollars.

When Zimbabwe gain independence in 1980, their dollar was about worth one United States dollar (USD). When my friend gave me the 100 trillion dollar note shown, it was worth about one USD.

Update 8-17-2021 - see this post for how pi has been calculated to over 62 trillion digits.

One Trillion!

I'm a regular blood donor and in the photo I'm shown with the local blood bank staff as I surpassed the 10-gallon milestone (we're all in masks since this donation was made during the Covid crisis of 2020). This particular session, I donated both blood and platelets. As I finished, I reviewed the monitor on the apheresis machine that summarized my donation: 980,000,000,000 platelets collected (just 2% shy of one trillion).

1/15/2022 Update - I recently reached my 15-gallon milestone. I checked my last platelet donation, and the count was 1.1 trillion.

Project Euler

In response to reading my blog, a friend suggested that I join Project Euler. The site has over 700 problems to challenge its members by solving mathematical problems using programming. The group just crossed over one million members this year. 

As a fan of Grant Sanderson's 3Blue1Brown site and his many videos, I took his suggestion of improving my mathematical skill by learning new programming. It's been over 25 years since I did any serious programming, so my skills are rusty and outdated. I've been learning Python and slowly I've been able to solve some of the Euler Project challenges.

Here's an example of a problem from Project Euler

Problem 9

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

a² + b² = c²

For example, 3² + 4² = 9 + 16 = 25 = 5².

There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.

I gravitated toward this problem because one of my first posts on this blog dealt with Pythagorean Triples. I used the tip posted to get a head start in finding the desired triplet of (200, 375, 425). Achieving the correct answer was good reinforcement of Grant Sanderson's advice to learn new programming skills.

How Rare is That Dollar?

The one-dollar bill shown in the photo is somewhat rare. Not so rare that it is worth more than one US dollar, but rare from the perspective of its serial number. That fact may seem surprising. At first glance, the serial number 29573801 doesn't appear extraordinary in any way; it looks just like one of the random 100,000,000 possible serial numbers possible in the 8-digit sequence. You'll see why it is rare after reading the following challenge.

In a prior post, I described a friendly bar bet in which you could easily estimate the number of prime numbers found in a range between two given numbers. A good friend was disappointed with the bar bet because most of his drinking friends don't know what a prime number is. So, I'm offering him a new challenge to try.

Ask your friend to take the first dollar bill out of his wallet or billfold. Without either of you looking at the serial number, he should place it on bar with his hand covering it. Now offer him $10 if the serial number has no repeated digits; if it doesn't, you win the bill. The bill in the photo would be a winner for your friend. Its serial number of 29573801 has no repeated digits. 

The wager is heavily weighted in favor of you. Only about 1 in 55 bills will have a serial number with no repeated digits. 

To see why this occurs so infrequently, consider the specific bill pictured in the photo. Of the digits 0 through 9, it is missing 4 and 6. Let's first calculate the probability that a given bill is missing the digits 4 and 6 and doesn't repeat any of the numbers. Moving through the serial number one place at a time, to meet this criteria, the first digit can be any number except 4 or 6. That leaves 8 digits out of 10. So the probability of getting through the first digit is 8/10. Now, moving to the second digit of the serial number, we need the probability that it is not a 4, 6 or a repeat of the first digit. That leaves 7 digits, so our probability of making it through the first two digits is 8/10 x 7/10. Continuing with all the remaining serial number's digits we find a probability of 8/10 x 7/10 x 6/104 x 5/10 x 4/10 x 3/10 x 2/10 x 1/10 which equals 40320/100,000,000 = .0004032. Next we have to adjust this probability because it was for the very specific case of a bill having no repeated digits and no 4 or 6. There are a total of 45 combinations of the two excluded digits, so our final probability is .000432 x 45 = 0.18144 or roughly 1 in 55.