# Robin the Frog Wasn't Halfway Down

## 'Halfway' and 'The Middle' aren't always the same thing.

There are thirteen stairs on the staircase at home.  In other words, if I stand on the landing and walk down the stairs counting one for each downward step, when I get to thirteen I am at ground level in the hallway.

So when am I halfway down the stairs?  This is no trivial matter, because Robin the muppet frog, Kermit’s nephew, sang a hit song called ‘Halfway Down the Stairs’ (based on the poem by A.A.Milne), in which he declared that he always liked to sit halfway down the stairs.

In our book Maths On The Go, one of the games we suggest that parents play with their children is called ‘Halfway Down the Stairs’.  In the first draft I wrote the following:

‘When you go down the stairs with your child, count how many steps there are.  Between you, work out when you are halfway.  You can discover together that if you have an even number of stairs (such as twelve) then there is an exact halfway stair (in this case number six – you can climb up six or go down six from there), but if you have an odd number of stairs then there isn’t a halfway stair.’

And since at home we have thirteen stairs, I reckon I don’t have a halfway stair.

I thought no more about it until a colleague read it.  He vehemently disagreed. ‘Your halfway stair is stair number 7’, he said.  His reasoning: if you stand on stair seven, then the stairs below you are stair numbers 1,2,3,4,5,6 and the stairs above you are 8,9,10,11,12,13. So number 7 is the middle stair.

Now I concede that the seventh joined sections of vertical and horizontal wood that make what we call a stair is the middle stair, but I argue that this is not the same as ‘halfway ’.  To me halfway up means one has gained half of the elevation required to get to the landing at the top of the stairs  (if one is climbing) or half the descent needed to reach the hallway at the bottom (if one is descending).

So I stand by my claim that if you have twelve stairs, then halfway up the stairs is stair six (because when you count six you are standing on top of the sixth stair).

What about Robin the frog? If you study the Youtube clip of his song carefully, you’ll see that by coincidence his staircase has 13 stairs, just like mine.  He is sitting on stair 7 yet he claims to be ‘halfway down’.  I protest – I don’t think he’s made it yet.  Another friend points out that his feet are actually just about on stair six, so he’s kind of split between stairs 6 and 7.  I argue that when you are sat on the stairs with your feet on the stair below, you can’t have it both ways, you’re either on stair A or on stair A+1, and with thirteen stairs then neither of those stairs is halfway up, nor is it halfway down.

Yet I am encountering some passionate disagreement. In fact based on my poll of adults across the nation so far, I'm in the minority.  About 60% think that stair 7 is halfway, while only about 25% agree with me that halfway is between stairs 6 and 7, and the rest think there is surely more to life than this.* Next thing you know, people will be arguing that dress in the photo was white and gold, not blue and black.

* A small number of people say 'It depends what you mean by halfway down'.  And they are right.