**From New Scientist #1403, 29th March 1984** [link]

“You’re looking fit!” I remarked to my nephew Henry.

“Thanks,” he replied. “I keep fit by running down Bill’s field then walking up.”

“Do you know your average speed?” I inquired.

“I’ll let you work it out — it’s a one digit number — while my speeds, up and down, are whole numbers. As you know, I don’t like fractions, and all these speeds are in miles per hour.”

“I don’t think you’ve given me enough information,” I complained.

“I think I will have, if I tell you the difference between my speeds, up and down the field.” Then he did so.

“Still not enough information.”

“Sorry! — I should have told you that I run down the field at less than 17 miles an hour. Now you have enough information.”

What was his average speed?

[enigma256]

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Solution:Henry’s average speed is 5 mph.Hi Naim, the program you have posted gives 5 possible solutions. How do you decide which one is the correct answer?

Hi Jim,

In my opinion, the nearest speed to 17 (run down) should be selected.

Is there enough information for a single solution?

Yes – there is enough information to get a single solution. A careful reading of the puzzle text gives enough information for a unique answer to be found.

If you need a more detailed explanation I’m happy to explain further.

I reasoned as follows:

Suppose he runs at speed r and walks at w (m.p.h.). His average over the double distance is 2rw/(r + w). For all values to be integral and the average less than 10, we could have

6 and 2: difference 4, average 3

6 and 3: difference 3, average 4

12 and 4: difference 8, average 6

12 and 6: difference 6, average 8

15 and 3: difference 12, average 5

18 and 6: difference 12, average 9

20 and 5: difference 15, average 8

28 and 4: difference 24, average 7

There was ambiguity, so the difference must be 12; then 18 was eliminated, leaving 15 and 3.

Yes, that’s right. For completeness you could also consider speeds of 45 mph (down) and 5 mph (up), which also give an average of 9 mph. Although according to Wikipedia the fastest recorded human footspeed is just under 28 mph (by Usain Bolt – [ http://en.wikipedia.org/wiki/Footspeed#Limits_of_speed ]), so it’s not really plausible.