Change Management

During a consulting interview, it is likely that a potential candidate will be given a brain teaser in order to assess the individual's ability to logically work through a problem.

With this in mind, I thought I'd attempt to unearth one of the most intriguing questions of all time and perhaps the greatest brain teaser ever given to a potential consultant candidate.

Some brain teasers include: Why are manhole covers round?, How many pennies would it take to reach the top of the Empire State Building?, How many golf balls fit into a Boeing 747? (Vault.com)

As you can see, the questions are quite abstract and probably would tell an interviewer a great deal about how that applicant works through ambiguous problems. But, after a while, these same brain teasers can become boring. Therefore, I wanted to work through my very own brain teaser that I find particularly interesting.

• How much time does the average person engage in sexual intercourse during their lifetime?

In order to answer this question we have to consider many things, but it will be necessary to break down one's lifespan into sizable chunks that can be dealt with in a logical way, due to the fact that people are engaging in more or less sexual activity depending on their age on average.

First, some background information. I am only going to figure this out for the United States population. Other countries may be having more or less sexual intercourse over a lifetime. The average lifespan for an individual living in the United States is 78.06 years according to Wikipedia. This ranks as tied for 30th place in the whole world. The life expectancy is slightly different for males and females, but for purposes of this brain teaser, we will just take the average age.

Now we have to consider when people actually start having sexual intercourse. According to a publication by American Sexual Behavior, an individual's first sexual experience will occur around the age of 17 years. I will then assume that once an individual starts having sexual intercourse they will be able to continue to do so. Therefore, there are 61.06 years of life in which sexual intercourse is possible on average.

However, the frequencies of sexual intercourse are different depending on the phase of life you are in. Consider the frequency of sexual intercourse individuals will typically have when first starting to have sexual intercourse versus when someone gets married versus when someone is a lot older.

To uncover the frequencies of sexual intercourse in one's life I drew research done from the 2005 Durex Global Sex Survey. The results are based on a responses from almost 317,000 people worldwide. This is considered one of the most comprehensive sex surveys in the world.

According to the survey results, people aged 16-20 are engaging in 90 sexual experiences a year compared with 108 sexual experiences for 25-34 year olds, but the highest frequency of sexual intercourse is occurring for individuals aged 35-44 years with 112 sexual experiences a year.

Based on this data, I constructed a graph that would express the average amount of times per year an individual was having sexual intercourse at each age between 0 and 78.

Looking at the graph, I want to explain a couple features. First, the solid blue line is based on factual data that was attained and the black line is a polynomial function that best fits the data provided. The blue line is at 0 until age 17, because this is the first time that people engage in sexual intercourse on average according to documented research. Then, I graph the levels attained by Durex's Sex Survey for ages 16-20, 25-34, and 35-44. According to Durex, 35-44 year olds are having the "most" sex. Therefore, I consider the amount of 112 sexual intercourse experiences a year as my maximum and projected the data after the age of 44.

In order to make these projections, I found out that "Among 60- to 70-year-olds with partners, 46 percent of men and 38 percent of women have sex at least once a week" (HealthyMe). Therefore, I assume that at age 65 individuals are having 50 sexual experiences a week on average, which is a decline of 2.95 sexual experiences per year from 45 years old to 65 years old.

I then assume that at age 78 there are no longer any opportunities for sexual intercourse, on average, and therefore find the rate of decline from 50 sexual experiences at age 65 to 0 sexual experiences at age 78. This is a decline of 3.85 sexual experiences per year, on average.

Using the Trendline tool in Excel, I fit the graph that I had constructed with a binomial function that could generally explain for the amount of sexual intercourse an individual has per year, on average, based on the research I had collected.

The binomial function is equal to:

Where x is equal to the age of the individual. Using this formula, I was able to figure out the amount of times per year an individual has sexual intercourse from age 17 through 78, on average, for each year.

In order to figure out how many times during a lifetime an individual will likely have sexual intercourse, I find the sum of the frequency of sexual intercourse episodes a year for each year from 17 through 78. This is expressed as:

Performing this calculation, I arrive at a total of 4835.05 instances of sexual intercourse over the average lifetime for the average individual in the United States.

But, I am not done yet. I now need to find out how much time this equals. According to the Durex Global Sex Survey, "Americans spend 35 minutes on foreplay and sexual intercourse each session" (Source).

Therefore, by simply multiplying 4835.05 by 35 minutes, I arrive at the total amount of time having sexual intercourse in minutes for the average American. The amount of time spent having sexual intercourse over the average lifetime is therefore: 169,226.8 minutes.

This value is equal to 2820.4 hours and 117.5 days. In terms of an individual's average lifetime, the amount of time having sexual intercourse only equates to 0.41% of total life activities, on average.

• Conclusion

In order to figure out the amount of time an individual has sexual intercourse over a lifetime in the United States on average, I found out when people start having sexual intercourse and what the average life expectancy is. This gives an average "window of opportunity" for sexual intercourse.

Then by researching the frequencies of sexual episodes per year for various age segments of the population, I was able to determine a binomial function that expresses the amount of sexual experiences had for each year during the "window of opportunity."

I then added these to arrive at the average amount of sexual intercourse experiences had in a lifetime. I found that it was approximately 4835 sexual intercourse experiences in an average American lifetime.

I then found out the average length of time a sexual intercourse experience lasts, 35 minutes, and multiplied this number by the amount of sexual intercourse experiences over an average lifetime.

This calculation gives us our answer to the brain teaser, "How much time does the average person engage in sexual intercourse during their lifetime?" The determined value is 169,226.8 minutes or 2820.4 hours or 117.5 days or 0.41% of the average person in the United States' life.

Bar Lines

Recently, I surveyed several college students to get a gauge of how significant the length and time involved in waiting in line to get into a bar is on people's preferences to go to that bar. I think that there are implications for bar owners and the people who work their doors in order to maximize positive feelings relating to their bar, even if only from the experience of waiting in line.

My findings suggest that people are most definitely waiting in line and have a threshold in terms of both people waiting in line and time devoted to waiting in line at which a person becomes deterred from the establishment entirely. This is obviously on the average and doesn't necessarily explain the behaviors of all individuals, everywhere.

• Results

The results of the survey are based on 82 respondents. The respondents varied in age from 19 to 27, with a mean of 21.6 years. The respondent's genders were evenly distributed (Males = 41, Female = 40, No Response = 1).

Generally, it appears as though the wait time in lines is generally pretty quick, however, there are some individuals who feel as though they spend lengthier amounts of time in line ranging from around 15 minutes to 25 minutes. These are quite significant time investments in order to experience the inside of a bar. There was no statistical finding that indicated that female respondents wait in line for a different length of time than male respondents, which is surprising.

Here are the results of one of the questions in the survey. It reveals that the respondent's in this survey are definitely influenced by the length of the line in terms of going to a bar. The majority of respondent's indicated that between 20 and 30 percent of the time they would not go to a bar purely based on the length. Several respondent's had even higher frequencies of this preference.

The preceding distribution of respondents is even further warranted by the finding that the length of a line outside of a bar is quite a deterrent to those individuals who would like to attend a bar. Taking a look at the results from this chart, one can see that a majority of respondents felt as though the line outside of a bar to be quite a large deterrent, measuring an 8 out of 10. A very small number of respondents felt as though the length of a line had a small degree of deterrence, heavily outweighed by larger degrees of deterrence.

The importance of the length of a line outside of a bar to an individual also tells an interesting story. Based on these results, it appears as though lines are generally quite important to individuals in evaluating which bar to go to. A high frequency of individuals had this ranked very high in importance, while other individuals had this ranked as only moderately high. However, it is important to note the lack of low importance that a bar line plays in the minds of individuals going to bars. It is clearly something that they think about.

The number of people also seem to have an influence on the willingness of an individual to wait in a line to get into a bar. It appears as though there is a definite threshold at which individuals become deterred from waiting in line based on the perception of how many people are in line. At the point the line reaches 45 people, individuals are just not willing to wait anymore in that line. While the majority of individuals begin to be deterred from a line that has 25 people in it, this is not yet the point at which people completely give up on the line. At 45 people, or at least the perception of 45 people, becomes far too daunting to the person waiting to get in line and he or she thus removes him or herself from it.

People also tend to go to bars mainly because they have friends that are there. As the results of the pie-chart above indicate, a large portion of people choose a bar based on the fact that they have friends there. Other competing elements to this are the atmosphere of the bar and drink specials there.

Finally, I looked into whether or not people were even inquiring about the length or size of a line outside of a bar before even getting to the bar. I feel like this is another suitable gauge of indicating how important the line outside of a bar is in determining an individuals choice and behavior. A large majority (70%) have inquired about the line ahead of time, therefore, suggesting that the line is something that people are thinking about even prior to actually arriving at the bar.

Taking all of these findings together, I propose some thoughts in my conclusion.

• Conclusion

The length and size of a line outside of a bar has significant influence on our preference to enter into that bar or even wait in line. There are several reasons that draw me to make this conclusion. First, people are inquiring about the size of a line to a bar, either by texting or calling friends already at the bar or in some cases asking cab drivers whose typical routes may take them to and from the bar, which suggests that this is important information in the process of personal decision making. Most people also rank the length of the line outside of a bar as a very important determinant in going to a bar.

Based on the frequency that individuals don't attend bars based on the length of the line alone, it is clear to see that bar line length ought to be considered by management of the bars.

Alternatively, it may be management's prerogative to keep a longer line outside of the bar in order to feign that the inside of the bar is very cool. This adds an element of exclusivity to the bar, which is a concept that I chose not to test in this current survey. Exclusivity is important, perhaps in bigger cities and when individuals have more money to spend, but in a college town, exclusivity loses its appeal the moment people have to wait in line for too long or there are too many people in line to begin with.

There is a tricky balance to be struck by people moderating lines outside of a bar and the choice of management in terms of how exclusive he or she would want the bar to be.

However, there can also be services that bars may want to provide to their customers in order to keep them in the loop as much as possible, seeming though the line is an important determinant for individuals attending bars. Bars could look into broadcasting the length of their line over the Internet and making it extremely accessible via iPhone or Blackberry so that the knowledge can be portable. This will help keep customers informed and content.

Do Not Raise Taxes: Letter to Stabenow and Levin

I have been watching a great deal of programming regarding the current economic crisis. I couldn't help but stop myself from sending this letter to the current Senators of the state of Michigan about what a mistake it would be to raise taxes on the richest portion (those individuals who earn higher than 250K a year) of the United States. It would be madness and President Obama would destroy the essence of this country. Please read the letter to further understand why this cannot happen.

Senator Levin,

I must admit, this is the first time I have ever written you. While there have been several incidents over the past couple years that have disturbed me to such a point that I have thought to seek out higher powers in order to alleviate their effect, I have decided that I finally must say something.

There is no way that taxes on the richest families (250K+) in the United States can occur as President Barack H. Obama has proposed.

I understand that Treasury Secretary Timothy Geithner has suggested that the tax increases are, "necessary to make a down payment on health care reform and to limit future budget deficits (AP, Key Democrats oppose Obama's tax deduction plan, http://news.yahoo.com/s/ap/20090305/ap_on_go_ca_st_pe/obama_taxes).
While I support health care reform, we must consider our priorities considering the current economic situation. The establishment of health care reform would be ideal, but it is not the time to do so, and President Obama and Congress need to understand this. Our economy, our livelihood, our way of life, and our stature as a Superpower in the world is at stake if our economy fails us.

We must prioritize our economy now. It is very sick. It is afflicted with what financial pundits refer to as a "Credit-Crunch." The system has stopped flowing. Money has no velocity.

Consider a river that doesn't flow. It is no longer a river, but a still-pond. We cannot allow our economy to go from a river to a still-pond; money must flow in order for the United States of America to function.

By taxing the richest portion of the United States, we are sucking more of that money out of the system. We are further "crunching" the worst Credit-Crunch that has ever occurred. These individuals will breathe life back into the Credit markets by helping the velocity of money. These individuals will be the first ones to go back to banks and get loans that will be wise investments for the banks themselves, because they are loans to intrinsically wealthy people. They will likely not default (what has led us to our current financial mess) and will thus spur the velocity of money and life back into the overall system. These are wise people to handle money for the United States economy. I trust them and you do too.

I am currently a student at the University of Michigan - Ann Arbor. I am a senior majoring in Organizational Studies and am weeks away from graduation. I work at the Michigan Telefund, in which I call alumni and student's parents soliciting donations to fund the University of Michigan's $7.5+ Billion endowment.
Soliciting donations from the American people today is laughable. People have been laid-off, are getting divorced, and don't have a cent to give to the University or anyone for that matter.

I told a parent tonight that I didn't choose the job because it was going to be easy, I chose it because I knew it was going to be hard.

Regardless, if people funding the education of students at the University of Michigan (where tuition is $47,000 a year) can't part with $25, who possibly can?

If we take more money away from the richest demographic of the United States, they will literally have nothing. We will destroy this country from the inside out as all great empires have done through the ages. We can error as they have, or we can learn from our mistakes.

Thank you,
Tyrone Schiff

Identity

Mathematics is a complex and beautiful science. It is a logical poetry that can be to some just as majestic as the writings of Shakespeare, Cummings, Emerson, or Blake. Several years ago I was introduced to this Identity riddle of sorts.

In mathematics, each value, whether it is an integer or a decimal or a fraction or anything really, has its own value that is unique to it. The number "5" is the number "5." There isn't a lot of argument in this. Surely you can depict the number 5 in several different ways, but there is only one value for that number. 5 is not equal to 7, nor is it equal to 0.673, nor is it equal to the Sin (60 Degrees).

By now, I hope you get the idea that numbers have their own identity. But the proof I am about to show you may alter your view of mathematics and perhaps even reality in general.

The sequence is quite simple really. I'm going to show you several variations of the same fraction and then I am going to suggest that if patterns hold, and in this case they most certainly do, then there is a discrepancy as the value approaches the number 1.

Enough of this abstract jibber-jabber. Let me work through the problem with you now. This problem involves the fractions and decimal values when a number is divided by 9.

When a number is divided by the value of 9, a really interesting pattern seems to emerge in the decimal value that results. I'm going to give you a couple examples and I'm going to see if you are able to decipher the pattern.

For instance, when the number one is divided by 9, the result in decimal form is .111 repeating. This means that until infinite the number 1 will reoccur. Visually it is depicted as follows:
Great. So, just so that everyone is up to speed, there are essentially two things that we learned from this one fraction. First, that the decimal reoccurs until infinite, and that the decimal number is the same number that is being divided by. In this case that would equal 1.

But, lets look at a couple other examples of numbers divided by 9 to see if there is an emerging pattern that we may be able to formulate.

Consider the fractions and decimal results from the number 4, 7, and 8:

As you can see, there is most definitely a pattern that is beginning to emerge when values are divided by the number 9. In all of these cases, the result is a recurring decimal until infinite and the number that is recurring is the same value as that being divided by. Excellent.

Well, with this in mind, that the number that is being divided by creates a decimal that is recurring until infinite while also being the same number as the value being divided by in the decimal, a minor problem arises when one considers the case of the number 9 divided by itself.

Any number divided by itself is 1. This is the result of the multiplicative identity which proves that all numbers that are divided by itself render the same number, 1.

However, this is slightly disconcerting given our current pattern involving numbers that are divided by 9. If we notice that 1 divided by 9 is 1 "repeating," and 4 divided by 9 is 4 "repeating," and 7 divided by 9 is 7 "repeating," then surely the same must hold for 9 when divided by 9 is 9 "repeating." According to the multiplicative identity this just isn't the case.

This strikes me as a little perplexing, and the desired effect of this blog post would be to elicit similar feelings in you. There is a definite conundrum here. There are two numbers that can be proven separate ways that equal the same thing, but are just not the same. In my opinion, this defies most of what mathematics is, a proof of why something is as it is. And though in math you can prove things several ways and arrive at the same result, the result here is different.

I hope you enjoy pondering this complexity as I have.