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#180 Neat as a PIN

I sometimes have difficulty remembering my five-digit PIN. I am cautious, so I don鈥檛 want to write it down, and I use five different digits so it isn鈥檛 easily guessed.

Fortunately, it has a special mathematical property that allows me to work it out if I ever need to.

All I need to remember is that if I add an extra digit to the start of the PIN, it makes a new number that is an exact multiple of the new number created if I add the same digit to the end of the PIN. As it happens, this multiple is also the same number as the extra added digit.

It is enough to defeat most people, but can you work out what my PIN is?

#179 Martian food

Solution

Let鈥檚 call the sustainable population of Mars 鈥淧鈥�.

P people consume the amount of grass that grows in any period. 40-P people consume the remaining stock in 20 days. 20-P people consume the remaining stock in 60 days.

So 20 x (40-P) = 60 x (20-P) and hence P = 10.

(This is a modern variant of a puzzle first set by Isaac Newton, which involved oxen grazing on pasture.)