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#210 Action station

Set by Mary Ellis

I have a train to catch! I was planning to drive or cycle to the station, but, to my dismay, I realise that my car has a flat battery and my bike has a puncture.

I look at the clock and realise that, based on past experience, if I were to set out right now and walk to the station, I would miss my train by 10 minutes. However, if I were to run at my top speed, I would be 15 minutes early. It is freezing outside and I don’t fancy waiting on the platform for any longer than I have to.

It is too late to do any more calculations as I need to leave right now. I decide to run for half the distance and walk the rest of the way. But where are my running shoes? After a frantic search, I find them. I leave the house 2 minutes later. Do I make it in time to catch the train?

Solution next week

#209 Postman’s knock

Solution

There are 257 houses in the road.

Experimentation will quickly establish that no two-digit number has the required property, so let’s find a three-digit number whose digits are a, b and c in some order. We know that multiplying a × b × c = 5(a + b + c), and since this product is a multiple of 5, one of a, b or c must be 5 (let’s say it is c). That means 5ab = 5(a + b + 5), and hence ab = (a + b + 5).

The values of a and b can be 2 and 7 or 3 and 4, meaning the house number could be 257, 275, 345 and so on. However, since only one house – the one at the end of the road – has this property, the number of houses on the road must be the smallest of these, which is 257.