The second Problem of the Week has now been posted. More of a puzzle this week, rather than a conventional math problem. Enjoy! I’ve also posted a solution to last week’s problem. Enjoy that too!

Gotta run now. Breaking Bad is on in just over two hours, and I have to begin my preparations.


  1. #1 Poincare
    September 8, 2013

    I notice that “Problems are available at the bulletin board outside Roop 119.” What is this new-fangled educational tool you have at GMU, called a Roop? They must be pretty good since you seem to have at least 119 of them.

  2. #2 Jason Rosenhouse
    September 9, 2013

    Ahem. We are JMU. GMU refers to our rivals George Mason University.

  3. #3 Poincare
    September 10, 2013

    OK JMU, what’s a little spelling among friends. You are my friend aren’t you?
    Now, what about “Roop”?

    I assume you would rather not have solutions posted online before Friday where contestants can see them, but I was able to get from
    A–X to BXX
    BXX…..A–X in 17 moves but it took 18 to get to


  4. #4 Divalent
    September 10, 2013

    A minor difficulty with the problem sheet you posted, with respect to the form of the answer you suggested: you didn’t name the other three items and their initial starting locations. (and the illustration is doesn’t allow one to easily identify what those objects are). So to solve the problem we first have to assign a name and location to those other 3.

    (hockey puck, mug, candle holder?)

    And I was able to to the exchange in 17 moves, except it left two of the other pieces in different spots than their starting locations. I.e.,:
    Started with

    Ended with

    So I solved the specific narrow problem posed (exchanging W and B), but I’m not sure whether the location of the other 3 items was supposed to be the same or not.

  5. #5 magster2
    September 10, 2013

    Divalent, I cheated and used a computer. Following your notation, it turns out it is impossible to reach the arrangement where W and B are switched and the remaining items are in their starting positions. (In fact half of the possible arrangements of items are unattainable.) There is also a second solution with 17 moves.

  6. #6 Jason Rosenhouse
    September 10, 2013

    All that is required is that the flask and the brush be transposed. It makes no difference where the other items end up. The other three items are a mousetrap, a pepperbox and an iron. But it’s certainly fine just to label them A-E and got from there.

  7. #7 Rick
    September 11, 2013

    I’ve been traveling and haven’t had a chance to really sit down and solve this one. Like others I first read it to intend that the other items should end up in the same locations. Bu t I knew from experience with similar puzzles that, in general,z half of the posible arrangements cannot be reached, including those in which the location of two items as switched and all the others remain in the same place. So I am not suprised that this first reading was incorrect.

  8. #8 magster2
    September 11, 2013

    Rick, I made the same incorrect first reading as well and proceeded to be driven crazy from attempting to do the impossible. The following (taken from http://www.jwstelly.org/CyclopediaOfPuzzles/PuzzlePage.php?puzzleid=Pz82.1) from what I assume is the original presentation of the puzzle gives a slightly stronger hint that the other items don’t need to end up in the same places by referencing them without imposing the limitation:

    “It is only necessary to mention that the whisky flask represents the bedstead and the scrubbing brush may be taken for the ice box, and that you are to transpose the positions of these two articles by moving one piece at a time in a sequence of plays in which the flat-iron, pepper box and mouse trap may be used to advantage.”

  9. #9 Sean T
    September 16, 2013


    I enjoy your POTW series, so I already looked at and solved (don’t worry, I won’t post it on your blog) your POTW 3. I do think, for the sake of completeness, though, that you need to edit the statement of the problem. Since I know the rules of chess, this was not problematic for me, but you should indicate in the statement of the problem that the queen may not move to a square adjacent to the king. That’s obvious to a chess player, but maybe not to a non-chess player. Without that restriction, the following is a valid solution, and I am sure that you do not want it to be one:

    1. Qb1+ Kd2
    2 Qc1+ Kd3

  10. #10 Peter
    September 18, 2013

    Sean T,
    I’ve searched the rules of chess and I cannot find any prohibition on the queen moving adjacent to the king.

  11. #11 Ken Adler
    September 19, 2013

    A little late to the conversation about lines but the comments over there are shut off. y=mx+b is not a linear function, it is called an affine function. To stay safe, call y=mx+b a linear equation.

  12. #12 Sean T
    September 19, 2013


    Okay, that’s not the actual rule in chess, but for the purpose of this problem it’s equivalent. The rules of chess allow the king to capture the queen if it moves to a square adjacent to the king. If this happens, obviously, the king cannot be forced to move to d3, and the problem cannot be solved. Therefore, disallowing a move by the queen to a square adjacent to the king eliminates the two-move solution I posted above.

    I know it seems nit-picky since it’s likely that most people reading the problem know enough about chess to know that the pieces can be captured. However, in the problem it is stated that no chess knowledge is needed to solve the problem. Whether you want to state that the queen cannot move to a square adjacent to the king or that the king is allowed to capture the queen if it can legally move to a square occupied by the queen is really not the point. The point is that there is a missing axiom in the statement of the problem. This missing axiom leads to an unintended (I am assuming) valid solution, assuming that you follow the rules as stated in the problem and not the actual rules of chess.

  13. #13 Jay Walker
    Fort Wayne, Indiana
    September 19, 2013

    Greetings Jason,

    My blog may be off subject but i am experiencing a very serious situation and i require you & others urgent assistance… I dont know exactly what to do….i’ll explain.

    I am a private practicing grassroots research scientist just over 18 yrs in the field of Socio-Economics and i am more than thrilled to inform you and others that i have made great discoveries & made sound developments in workforce development and economic upgrades that will accurately aid everyone in the areas where needed all that is required is certification of my work….

    ….but my problems lies in the area of non-scientific individuals in leadership here in my hometown of fort wayne indiana that made it into leadership by the use of our countries “out-dated public acquiring leaders etc. election system”. These non-scientific leaders are turly in the class of warriors and politically trying to publiclly destroy myself and work that our nation has been waiting on to aid with ending our economic crisis yet they are oblivious to what they are doing.

    I believe they are competing with me when i am only trying to supply them with the master socio-economic methods (business model) that will show to aid our city and nation with our economic mayhem. For the last four years these individuals have caused me great problems when i first tried to make my city-wide debut. They have gotten the local BBB and 2 of our news syndications to slander me locally.

    i sued both the BBB and news center and the president of their board for slander. The news center settled out of court and got rid of the host and his show directly after the signing of the settlement agreement but have refused to clear my name or pay me for damage done.

    long story short, the judges in the small claims court has been on both of their sides. My problems started when i turned down being on their board after they requested me to join twice. I denyed them due to the fact that i could see their plans and refuse to take part in robbing our citizens.

    I’m calling on the scientific community to aid a fellow scientist who will represent the intellectual society well once my breakthrough economic work and self is recognized.

    I’m calling on all sceintist to please help me…they are literally trying to run me into the ground by telling leaders in the city to not aid me in this breakthrough development that will show to accurately aid in ending our economic problems ….that are really only caused by non-qualified people in the areas of leadership or the positions only scientific community oriented intellectuals should be able to attain.

    this is a true citizens emergency…i’ve seemingly have been black balled and major church pastors here are also deeply involved.

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