The Wotsit Problem

Here in the Hive we firmly believe there is no problem we cannot tackle when we combine our powers and pool our genius. Yesterday afternoon stretched our powers to the limit as we pondered exactly how we could supply a paddling pool full of wotsits at our party this coming weekend.

You may ask "Why on earth do you wish to do such a thing ?" which would be a fair question really. The answer lies with Geoff's friend Dave who replied to the email invite with the following:

“Anyway, afraid your invite was 2 days too late, and I kind of said I'd go to another firework pi$$up down here. Gonna depend on how much in the way of munchies is being provided. . . . A paddling pool full of cheesy wotsits could influence my decision! “

Geoff was left with no choice other than to provide such a treat in order to lure Dave to the party at the Hive.

Of course this leaves a number of problems:

A) Procuring a paddling pool - a task and expense I am happily leaving to Geoff
B) The placement of such a pool - our house is going to be filled to the rafters as is and space is already at a premium
C) How many wotsits will be needed to fill the pool ?

Tuesday afternoon was devoted to sufficiently answering question c) Geoff kindly sent everyone an email containing his basic workings and assumptions for the task, which quickly captured the imagination of those who received it.

"The principal one is this: just how many Wotsits does it take to fill a paddling pool? I devoted my lunchtime today to answering this question.

My first port of call was the newsagents just down the road. To my dismay, they only stocked a new limited edition variation called “Twisty Wotsits”. Since use of these would foul up the calculations I quickly went on my way, cursing their ill-considered stock choices. I walked a bit further and found a shop that sold conventional Wotsits. On the way back to work, I started to consider the problem in earnest.

A paddling pool can be considered to be a short cylinder of relatively large diameter. The formula for cylinder volume is ∏r2 x h where r is the radius of the cylinder’s cross section and h is its height.

If we assume a cylinder radius of 0.75m and a height of 0.4m this gives paddling pool area of: 3.1415 x (75)2 x 40 = 706837.5cm3

A Wotsit can also be considered to be a cylinder (mathematically rigorous types may wish to point out at this stage that Wotsits often feature a slight curve which would no doubt complicate these volume calculations. For the purposes of simplicity we will allow the approximation of Wotsits to cylindrical shape, irrespective of their true geometry).

I found the “average” Wotsit by finding a Wotsit that seemed larger than its fellows, and another Wotsit that appeared somewhat runtish in comparison to its lengthier bagmates. I then found a Wotsit that was smaller than the large Wotsit yet larger than the small Wotsit and deemed this to be the average Wotsit. I measured this Wotsit and found it to be 4.5cm in length by 1cm in diameter.

Feeding this into the formula, this gives: 3.1415 x (0.5)2 x 4.5 = 3.5cm3
Now it is a simple matter of feeding in the numbers. 706837.5 / 3.5 = 201953.6 Wotsits required to fill the paddling pool.

Now, 201953.6 sounds like a big number (and it is), but when you consider that there may be as many as 37 Wotsits in a bag, this equates to: 201953.6 / 37 = 5458.2 bags of Wotsits required.

5458.2 is still a very big number in terms of bags of Wotsits. The final factor to account for is the space between the Wotsits – the preceding calculations assume that the Wotsits are laid out neatly with no space between them. As anyone who has ever poured Wotsits into a bowl will know, this configuration is unlikely to occur by chance. It is more reasonable to believe that the Wotsits will form an intricate and tangled lattice of deliciously cheesy puffed maize snacks. Therefore, we should factor in that 50% of the volume of the pool will be accounted for by space between the Wotsits.

This reduces the number of bags required to a paltry 2729.1, or 272 multipacks of 10. Good grief."

Clearly obtaining the best part of 300 bags of wotsits is no small feat or expense. Within half an hour or so Geoff received another email from a friend of his - Dr Smith. He gave the following advice:

"Geoff,
I have tested your elementary Mathematics...
We’d normally charge this type of consultancy at £££’s an hour, but I’ll waive the fee this time...

To give a ‘worst case’ figure I have ignored your 50% air figure and made an assumption that these wotsits will tessellate perfectly.
This would indicate that a single covering of the (rather small) paddling pool would require 13,783 wotsits.

At 4.5cm in height, you can then fit 8 full layers of wotsits to avoid overflow... giving a total of 110,264 wotsits.

At 37 per pack, that is 2980 packets... 298 (x10 multipacks)
a figure within 8% or 9% of your initial estimate... very good!"

The best part of 3000 packets of wotsits was not what we wanted to hear. Dr Kirk a good friend of mine also chipped in with his workings on the 'Wotsit Problem'

“Geoff your maths calculations are correct on the Wotsit problem. Also by some freak of nature if the calculation is done on a scientific calculator with a value of pi to 30 decimal places and all the calculations are done to the same d.p the answer comes to exactly 200000 wotsits, in fact the proof is show below so you can try it for yourself.

Area of pool = 706858.34705770347865409476123789 cm3

Area of wotsit = 3.5342917352885173932704738061894 cm3

Total number of wotsits = 200000"

200,000 wotsits or equivalent to 5405 bags ! Wow. It occurred to me as I wrote this that we have no idea what size the pool is going to be. I don't believe Geoff has found one yet or has one in mind, but I could be mistaken. Hopefully we'll end up with something relatively small and therefore need less wotsits to fill it.

One of the last things we received was an artists impression of Dave in the paddling pool from Bones :


The observant among you will spot the Rock's head in this picture. This is due to the uncanny resemblance between Dave and the Rock.

Lastly we received this from Mr Lichfield:

“Something I don't think the mathematicians have thought of or perhaps they did and I'd stopped paying attention. Anyway, I imagined Dave spending the evening sat in the paddling pool for ease of consumption, this would therefore displace a number of wotsits.”

It remains to be seen how this will pan out and whether we will succeed in filling a paddling pool with wotsits. Rest assured pictures will follow shortly afterwards if we manage it !