Puzzle: ant and honey

The shortest distance between two points on a plane is a straight line. But finding the shortest distance on other surfaces is a more interesting problem.

Here is a puzzle that is harder than it sounds:

In a rectangular box, with length 30 inches and height and width 12 inches, an ant is located on the middle of one side 1 inch from the bottom of the box.

There is a drop of honey at the opposite side of the box, on the middle of one side, 1 inch from the top.

What is the shortest distance the ant would need to crawl to get the honey?

Here is a picture that illustrates the position of the ant and the honey.

Can you solve it?

The answer is posted in the comments section.



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  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    The answer

    If the ant crawls 1 inch down, then 30 inches across the bottom, then 11 inches up, it will travel 42 inches. But this is not the shortest distance.

    The solution is found by unfolding the box and then finding the shortest path between the ant and the honey.

    There are actually 4 ways to “flatten” the box (shown here). But only one method corresponds to the the shortest distance as follows:

    The distance between the points can be found using the Pythagorean theorem. For a triangle with legs 32 and 24, the hypotenuse–and shortest distance–is 40 inches.

  • http://www.politicomix.net Roberto

    Superb.

    My guess would have been to have the ant travel horizontally to the “side” plane, then up the hypotenuse of the 30-inch side from one inch above the bottom of the box to one inch below the top (i.e. parallel to the honey drop), and then horizontally again to the honey. That is: 6 + 6 + 31.62 (square root of 1,000).

    I guess I don’t get that job at Google … ;-)

  • http://worksheet.budgibson.com Bud Gibson

    I like your solution, and it’s a clever problem. However, it’s important to note that your drawing represents only one interpretation of what’s written. For instance, nothing precludes interpretation of top and bottom as applying to two of the 30X12 sides. In this case, the distance seems to be either 24 or 14 inches.

    The 24 inch solution is based on the notion that top and bottom must make sense at the same time (no box rotation allowed). 1+12+11=24

    The 14 inch solutions assumes you can rotate the box so that your interpretation of top and bottom lead the two dots to be located within an inch of of the same side 1+12+1=14.

  • http://www.vpsgraphics.com V Paul Smith Jr

    I get what you’re saying, but there was no rotation mentioned nor implied, and it completely goes against any reasonable interpretation of what is written. Your suggestion would be like saying that “a penny laying on the table has heads on the top, and tails on top. Oh yeah, I flipped the penny half way through that sentence.” See, that is just stupid.
    If rotation was mentioned, but just not in a specific manner, then your solution would hold water. But otherwise, I’d say not.

  • john

    @bud, I agree with Paul with respect to your 14 inch solution. But your 24 inch interpretation would be valid if it weren’t for the picture. It would have been better to use the word “ends” rather than “sides” in the discription.

    I do think the problem should make it clear that the box is floating in mid air. If it were resting on something then at least part of the bottom would be inaccessible. And if the area around the used portion of the bottom was not available, the 42 inches over the top would be shortest.

  • http://worksheet.budgibson.com Bud Gibson

    I’m willing to concede that the 14 inch solution stretches things.

    However, I’m not sure I should accept the drawing as more than one possible interpretation of the wording of the problem. The key to the solution is in fact one of interpretation of what’s possible.

    If you do understand the drawing as part of the problem definition, then the solution proffered is definitive. My solutions were based on different possible interpretations of the words and context.

  • http://vpsgraphics.com Paul Smith

    Um…no. That is not an acceptable interpretation by any reasonable standards. Even many unreasonable standards. You see to agree with everybody else on what “opposite” means so we are good there. Why is it not clear to you what “top” and “bottom” mean? The ant is 1 inch from one an the honey is 1 inch from the other.
    There exactly zero mention of rotation in the description, so it us completely unacceptable to arbitrarily introduce said transformation, particularly right in the middle of the assignment if te top/bottom labels.
    Going that route, you could rotate the box such ant-honey is separated by 2 inches (both 1 inch from a common point on any given edge).
    Or one could work it so that they are separated by zero inches (occupying the same spot).
    If you agree that those are absurd interpretations, then you have to concede that yours suffers from the same flawed reasoning.

  • http://www.vpsgraphics.com V Paul Smith Jr

    Wow, major spelling grammar errors in that last post! I will refrain from replying via small phone input box while on the go.
    “You seem to agree…”
    …and the honey…”
    “There is exactly…”
    “…so it is…”
    “…the assignment of the…”
    “…such that ant-honey…”
    Sorry everybody for such a post.

  • Chandan Kumar

    There are two shortest ways for the ant to get to the honey drop and both the distances are 42 inches.
    1. Ant will crawl don one inch, then crawl 30 inch across the length to get to the other side of the box and finally crawl up 11 inch to reach to the honey drop.
    2. First crawl up 11 inch then crawl 30 inch to get to the other side of the box and then crawl down one inch to get to the honey drop.

    These two are the shortest path for the ant. Any deviation from this path will make the total distance more than 42 inch.

  • john

    @Chandan Kumar, check Presh’s solution (first comment) and you’ll see that 40″ is possible.

    @Paul Smith, your comments only address Bud’s 14″ solution. He has already conceded that was a stretch. The 24″ solution has the ant 1″ from the bottom and the honey 1″ from the top. they’re just not on the 12″X12″ ends, but on the 12″X30″ sides.

    The discription did use the ambiguous word “side” instead of the less ambiguous “end”. So, without the picture, 24″ would be the solution to the most reasonable interpretation.

    However,
    @Bud Gibson, The picture is preceded by the sentence “Here is a picture that illustrates the position of the ant and the honey.” It says “THE” position, not “a possible”, or even “a” or “one”. I have a hard time seeing how this can be viewed as anything other than part of the description of the problem (unlike the misleading “teasers” the myth busters sometimes used – like for their airplane on a treadmill segment).

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