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Programming Enigma Puzzles

28 October 2013

Posted by on **From New Scientist #1284, 17th December 1981** [link]

I am at

X, mounted on my velocipede, and thirsty: so I want to reach the river, which runs straight alongBYC, as soon as I can. I have to keep within three fields on the plan. In each of them I can maintain a steady speed, at 30 mph withinXBYZ(including the boundary), and at rates which I know (but you don’t yet) inAXZDandZYCD(including their boundaries).If I tell you that there are precisely three different quickest routes from

Xto the river, that should enable you to work out:(a) my speed in

AXZD,

(b) my speed inZYCD.

**Note:** In **New Scientist #1294** (with **Enigma 149**) an apology was published stating that this puzzle does not actually have a solution.

[enigma140]

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The apology published in New Scientist #1294 is as follows:

However if we restrict the route to have no more than 2 components I think we can solve the puzzle as the setter intended.

The most direct route, along XB, is entirely in field XBYZ, and is 3 miles at 30 mph (= 2 minutes per mile), so takes 6 minutes.

The shortest route entirely through fields AXZD (at speed

aminutes per mile) and ZYCD (at speedbminutes per mile) is XZ + ZY, and will also take 6 minutes. Hence:So, the remaining six minute route must be to cut across field XBYZ to a point on ZY (E), and then follow the remainder of EY through field ZYCD. This also takes 6 minutes, so:

squaring both sides:

and rewriting as a quadratic equation in terms of e:

This will have exactly one solution when “b² = 4ac”, i.e.:

substituting for 2a:

So either: b = 2, a = 0 (not possible), or: b = 10/13, a = 24/13, e = 5/6.

Converting from minutes per mile to miles per hour (

xminutes per mile = 60/xmph) we get:a = 32.5 mph, b = 78 mph.

Solution:(a) Your speed in AXZD is 32.5 mph; (b) Your speed in ZYCD is 78 mph.I think your suggested approach is an excellent way to save the enigma. I am concerned however with the wording of the enigma that says the speed is constant within any area “including the boundary.” Then what is the speed at a shared boundary?

I took the view that on a boundary you travel at the fastest speed available from the areas on either side of the boundary. Although I’m really just working backwards from the desired answer to find the conditions that make the puzzle solvable.

Really the western and northern fields are superfluous. My suggested rewording of the puzzle would be to have just the big field, with a hedge or fence along side XZ and a gap at X that gives access to a track immediately to the left of it. That track leads to a tarred road ZY — it has to be surfaced to allow such a high speed. There is no fence along that north side of the field. At 3 miles by 2 it’s more a plain than a field.

Perhaps it’s too much to expect mathematical puzzles to reflect the real world!