## Puzzle Day brain teasers

At the Ri we love a good puzzle, and to celebrate Puzzle Day we take a look back at some of our favourites from last year.

At the Ri, we love puzzles, especially ones involving maths, science and a bit of logic.

The early birds amongst you who tuned into BBC Radio 4's Today programme may recently have heard some Puzzles for Today set by our very own Shaun Fitzgerald.

As today is Puzzle Day, it seems like a perfect time for us to share with you some of our brain teaser highlights from last year.

### Puzzle 1 – Life expectancy

According to the Office for National Statistics, life expectancy used to increase by roughly two months a year, every year. But for various reasons, this growth has now stalled.

Based on this report, we posed the following question:

If life expectancy for men is currently 79 years, and women is 83 years, and assuming we are able to return to increasing life expectancy by two months each year, what would be the life expectancy of someone born today?

### Puzzle 2 – Bitcoins

According to this BBC news story, the price of a Bitcoin, which is notoriously volatile, has fallen to under \$5000 for the first time since October 2017.

Which inspired this question:

If a Bitcoin’s value varies from \$10,000 to \$5000 each month, and you invest \$10,000 each month. What is the average price of each Bitcoin you buy?

### Puzzle 3 – Fertility rates

In 2017, Cyprus had one of the lowest average fertility rates in the world, at one child per woman. We also know that Cyprus has a life expectancy of about 80 years.

If we assume that the population has an even distribution of age and gender, and that there is no migration, how long will it take for Cyprus’s population to decrease by 10%?

So, have you solved them yet?

If you want to know how we worked them out, keep scrolling down for the answers…

But no cheating!

### Solution 1 – Life expectancy

We’re going to assume that if you are an 83 year old woman alive today, your life expectancy is 83.

And for a girl born today, her life expectancy would increase by two months for each year she will be alive: 83 + y*2/12 (where y = life expectancy, i.e. the number of years she will be alive).

Which gives you this equation: y = 83 + y*2/12

And if you solve this for ‘y’ you get: y = 83 / (1 – 2/12)

So ‘y’ = 99.6 – almost 100 years!

And similarly for men, who currently have a life expectancy of 79 years:

y = 79 + y*2/12

Which gives a life expectancy of 94.8 years.

Read this Puzzle for Today on the BBC website

### Solution 2 – Bitcoins

Using a bit of logic we can see that on average, half the time you will buy a Bitcoin for \$10,000, the other half of the time you buy two Bitcoins at \$5000 each.

Which means on average, after 2 months you have spent \$2000, and will have 3 Bitcoins.

And so the average price is actually \$2000/3 = \$6667

Which is significantly less than \$7500, the average price of a Bitcoin ((\$10,000 – \$5000)/2).

Read this Puzzle for Today on the BBC website

### Solution 3 – Fertility rates

If we let the current population of Cyprus = X

This means that the number of women in Cyprus = X/2

If on average, each woman has 1/80th of a child per year, so by the time they are 80, they have had a full child, this means that the average number of births per year (B) is:

B = (X/2)*(1/80) = X/160

Then let the number of deaths per year = D where, given an even distribution of age, 1/80th of the population will reach 80 and die each year:

D = X/80

The net change in population each year must be:

B – D = X/160 – X/80 = -X/160

A decrease of 10% in the population is -10%*X, and n is the number of years it takes to see this reduction:

-10%*X=Xn/160

n=16 years

Read this Puzzle for Today on the BBC website

So, how did you do?

Did you get them all right, or did you take a sneaky peek at the answers to help you along?

Either way, we hope you're not feeling too much like our face-palming friend from the archives.

From us all at the Ri, we wish you a Happy Puzzle Day!

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