Maths puzzles in yesterday's Evening Standard

Question One

Public sector workers are paid almost eight per cent more than private sector workers - and the gap is widening, according to official statistics... TUC General Secretary Brendan Barber said: "Hourly pay has been higher in the public sector for over 30 years. This is because public sector workers tend to be more highly skilled doing jobs like teaching and health care."

Question - out of seven or eight million people on the public sector payroll (directly or indirectly), how many are actually teachers, nurses or doctors?

Clue: about three-quarters of a million.

Question Two

Diamorphine - pure heroin - has been given to addicts in a third of London boroughs over the past three years. They take it under medical supervision several times a day. The annual NHS bill per addict is put at £14,000.

Supporters of prescribed heroin say this is dwarfed by the cost of crime users might otherwise commit to fund their addiction. Experts estimate an addict spends £45,000 a year on average on street heroin.

Question: How many junkies inject heroin "several times a day"? I though once or twice was normal.

Supplementary: An ampule of heroin costs tuppence ha'penny. The other £38 cost per day is presumably "several" times a made-up figure. Why don't they just give the addicts a prescription for a couple of ampules a day and let them collect them from the dispensing chemist?

Question Three:

When Lehman failed, banks refused to lend or even transact with each other for a period, threatening a collapse of the entire banking system. On average, more than £250 billion changes hands between UK banks in so-called large payments every day. The system handles more than 40 times the UK's gross domestic product each year, but individuals encounter it rarely, perhaps only when they buy a house.

Just 18 banks are members of the real-time settlement system CHAPS which ensures that large payments are completed immediately. Salmon called it one of the "most systemically important payment systems in the UK". But the vast majority of banks - running to several hundred - choose instead to route their large payments through a member of CHAPS and so do not receive settlement until the end of the trading day.

That effectively means they are taking out enormous, unsecured loans with the 18 CHAPS banks. More than 50% of CHAPS payments are made by the 18 correspondent banks on behalf of customer banks, leading to potentially very high risks. If more banks joined CHAPS the risk would be spread more broadly.

Question: If all the cash machines physically stopped working, because of a technical failure, would that lead to a 'collapse' of the banking system? Nope. It would be a bloody nuisance but not the end of the world. Ditto if the inter-bank payments system grinds to a halt.

Supplementary: What sort of idiot multiplies an unfeasibly large number like £250 billion by the number of days in a year? Why not multiply the number of grains of sand on a beach by the number of MPs in the UK

Surely the point is that these transactions net off to about 99.99%, i.e. by the end of the year Barclays will have paid Lloyds £10,000 billion and Lloyds will have paid Barclays £9,999 billion, so will end the year owing Barclays £1 billion more than it did at the start.

Supplementary: Why didn't they (or why don't they) just run these payments through the Bank of England as intermediary to eliminate counter-party risk? There's no need for the BoE to actually ever make any payments, it can just chalk them all up in a spreadsheet and then sort out any accumulated differences at the end of each month or each year (i.e. in the example above, at the end of the year Lloyds would have to pay Barclays £1 billion).