Blue Monday

This coming Monday is ‘Blue Monday’, the most depressing day of the year. On this day, twenty individual people walk on the sidewalk of a straight street. They all walk in one of two directions: left or right. However, if they encounter another, they immediately turn around and walk the other way. After all, they don’t feel like meeting others on this tough day. If everyone walks equally fast and needs ten minutes to walk the entire street at this speed, how long does it take before they have all left the street? You may assume that reversing does not result in a loss of time and that each of the pedestrians starts at any point in the street.

 

Answer

This takes a maximum of ten minutes because that is the time it takes to walk through the street.

When two pedestrians meet, they immediately turn around and walk the other way. But if all pedestrians are the same as they are for this puzzle, then that is exactly the same as the situation in which they walk straight through each other and each continues in the same direction. If you look at it this way, you see that the maximum time it takes for all pedestrians to get out of the street is simply the time it takes for one pedestrian to walk the entire street.

 

Context

If you try to solve this puzzle by calculating the time it takes one specific walker to get out of the street, then it is a very difficult puzzle. Since you probably calculate how long it takes for this walker to meet another walker, how far he will have to walk once he has turned around and you will immediately again have to deal with the chance that he will bump into other walkers. From that point of view, it is a very difficult story.

However, if you take a step back and see ‘the big picture’, it is actually a very simple problem. As soon as the realization comes that two pedestrians who meet and turn around are the same as two pedestrians walking through each other, resolving the issue is child’s play.

Insights of this kind are more common, for example in physics. The flow of a fluid is easy to describe if we look at the fluid as a whole. But if you try to describe that flow by the movements of all the individual molecules that swirl in the liquid, you have a very tough job. As a data scientist, you also encounter this phenomenon frequently. In data analysis research, for example, you often see data points that you do not fully understand at first glance, but that are easy to interpret if you place them in the context of the situation in which they occur.

This puzzle emphasizes the importance of the bigger picture. And that is an insight that can be applied more often than you may think.

Leave the Office Early Day

‘Leave the office early’ day was created to make people aware that productivity decreases with more working hours. Let’s do a calculation. Assume that employees of a company may work a maximum of 12 hours a day, 5 days a week. In addition, they work the same number of hours every day. They start with 100% productivity on Monday morning. Their productivity decreases 10% per hour, so in the first hour they are still 100% productive, the second hour 90%, the third 90% * (100% -10%) = 81%. With a night’s sleep, their productivity doubles again with a maximum of 100%.

 

How many hours a day should employees work to achieve optimum productivity during the week?

 

Answer

The answer is 8 hours a day.

 

Explanation

There are 12 options, you can work 1, 2, 3, …, 12 hours a day. With one hour a day you start on Monday with 100% productivity, on Tuesday you will have another 100% productive time (productivity is doubled, but with a max of 100%). You work productively for a total of 5 hours.

When working 12 hours a day, you work 100% productively the first hour on Monday, the second 90%, the third 90% * 90% = 81%, the fourth 81% * 90% = 73%, up to and including the 12th hour: 31%. If you add this together, you work productively 7.2 hours out of 12 on Mondays. Because you end Monday with 31%, and this productivity is doubled at night, you start Tuesday with only 63% productivity. And then your productivity decreases every hour. For a working week with 12 hours, it looks like this:

Hour	day 1	day 2	day 3	day 4	day 5
1	100%	63%	39%	25%	16%
2	90%	56%	35%	22%	14%
3	81%	51%	32%	20%	13%
4	73%	46%	29%	18%	11%
5	66%	41%	26%	16%	10%
6	59%	37%	23%	15%	9%
7	53%	33%	21%	13%	8%
8	48%	30%	19%	12%	7%
9	43%	27%	17%	11%	7%
10	39%	24%	15%	10%	6%
11	35%	22%	14%	9%	5%
12	31%	20%	12%	8%	5%
Totaal	7,2	4,5	2,8	1,8	1,1

Total of all days: 17.4 hours productive (out of 12 * 5 = 60 hours!)

This way you can do this for all options. The optimum is then on an 8-hour working day, with a total productivity of 26.1 hours (out of 40).

 

Background

This is a typical example of an optimization problem. In this problem, we try to maximize productivity under a number of conditions. We see that both the minimum number of working hours per day and the maximum number of working hours per day are not optimal. The optimum is somewhere in the middle. This is because two things must be considered: on the one hand, your productivity in an hour that you are not working is of course 0, on the other hand, an employee recovers more slowly if the productivity is very low at the end of the day (double of something low is still low).

We often see optimization problems among our customers. A classic example is campaigns in which it must be decided which of your customers you are trying to reach (or not). It must then be considered whether the costs (campaign) actually outweigh the benefits (chance that someone actually accepts the offer). You then optimize which part of your customers will or will not send you a campaign.