## Valentine's Day puzzle

Nothing says romance like a good maths puzzle, and we have a particularly love-ly brainteaser for you here.

Love is complicated and so are puzzles – but unlike love, at least puzzles can be solved with a little bit of logic and mathematics. Our very own Shaun Fitzgerald drew inspiration from Valentine’s Day to put his latest Puzzle for Today to listeners of BBC Radio 4’s Today programme.

## How many roses are too many?

It is 14 February and Justin is sending his Valentine a rose.

He knows this is true love, so he plans on doing the same again tomorrow, but this time he’ll send two roses. And then he thinks, why not three roses the day after that?

Plans get out of hand, and Justin contempates continuing this pattern for a whole year.

Assuming it isn’t a leap year, how many roses does Justin need to buy for a whole year?

• Credit: Tuân Nguyễn Minh via Unsplash

## Solution to the Roses Puzzle

This question can be solved as an algebraic series. The sum of the number of roses over ’n’ days is given by:

n ( r1 + rn ) / 2

Where ‘r1’ is the number of roses on the first day (1 rose), and ‘rn’ is the number of roses on the nth day.

The number of roses Justin sends on the first day is always 1, and ‘rn’ will always be the same value as ’n’.

For instance if Justin sent roses for 4 days, the number of days ’n’ is 4, and the number of roses sent on that 4th day ‘rn’ is also 4.

So we can simplify the equation to be:

n ( 1 + n ) / 2

Or:

n ( n + 1 ) / 2

But where does the ‘divided by 2’ come from? This visual proof might help you understand:

We need the ‘/ 2’ term because Justin’s pattern of adding one rose per day to the number he is sending makes the sequence of daily cumulative totals a sequence of triangular numbers.

Think of it like a 2D pyramid - at the top you have 1 rose, on the next level down you have 2 roses, on the next level down you have 3 roses etc. creating a triangular stack of roses.

The easiest way to calculate the number of roses in that triangle is to see it as one half of a whole rectangle of roses. To get the whole rectangle we left-justify the triangular arrangement, copy it and rotate it to fill the rectangular area.

We can then easily work out the rectangle’s area – the total number of roses – by multiplying the height ’n’ by the width ’n + 1’.

Now we just have to divide this number by 2 to get the area of the original triangular arrangement of roses.

This gives us the equation:

n ( n + 1 ) / 2

There are 365 days in a normal non-leap year, so let’s plug this ’n’ value into the equation:

365 ( 365 + 1 ) / 2 = 66,795

So over the course of Justin’s romance-fuelled year he will send 66,795 roses to his Valentine.

A little too much? Perhaps. Good luck Justin…

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