how to solve this?

amishera · 2009-08-13 12:24:11

Hi,
how to solve this puzzle:

A customer at a 7-11 store selected four items to buy, and was told that the cost was $7.11. He was curious that the cost was the same as the store name, so he enquired as to how the figure was derived. The clerk said that he had simply multiplied the prices of the four individual items. The customer protested that the four prices should have been ADDED, not MULTIPLIED. The clerk said that that was OK with him, but, the result was still the same: exactly $7.11. What were the four prices?

One way to solve this problem that comes immediately to mind is to factorize 711000 using some integer factorization rules. Is this the correct method or is there any smarter way to solve this quickly and without less work? Also here we know that we can convert the dollar amount to cents and which must be integer. But in general the quantity under consideration(7.11) might be any fraction (like electric current or amount of water) in this case is it possible to do factorization of fractions?

Thanks.

mathsyperson · 2009-08-13 14:05:32

Factorising the answer is how I'd start, although if we're working with cents then we'd need four numbers that add to 711 and multiply to 711000000.

That thing factorises to give 2^6 x 3^2 x 5^6 x 79.
There are (I think) 11760 different combinations there to try, so if logic fails it can be brute-forced at least.

Maybe the fact that the four numbers add to xx1 can help restrict things.

Edit: Using that as a start point and a few minutes of scribbles, I've found an (the?) answer.

Ricky · 2009-08-13 15:09:47

That is the only solution.

For[a2 = 0, a2 <= 6, a2++,
  For[a3 = 0, a3 <= 2, a3++,
    For[a5 = 0, a5 <= 6, a5++,
      For[b2 = 0, b2 <= 6 - a2, b2++,
        For[b3 = 0, b3 <= 2 - a3, b3++,
          For[b5 = 0, b5 <= 6 - a5, b5++,
            For[c2 = 0, c2 <= 6 - a2 - b2, c2++,
              For[c3 = 0, c3 <= 2 - a3 - b3, c3++,
                For[c5 = 0, c5 <= 6 - a5 - b5, c5++,
                  d2 = 6 - a2 - b2 - c2;
                  d3 = 2 - a3 - b3 - c3;
                  d5 = 6 - a5 - b5 - c5;
                  
                  If[2^a2*3^a3*5^a5 + 2^b2*3^b3*5^b5 + 
                    2^c2*3^c3*5^c5 + 79*2^d2*3^d3*5^d5 == 711, 
                   Print[{a2, a3, a5, b2, b3, b5, c2, c3, c5}]];
                  ];
                ];
              ];
            ];
          ];
        ];
      ];
    ];
  ];

Ricky · 2009-08-13 18:06:10

I have a deductive argument proving that there is only one solution (and finding it). Unfortunately it's about 6 pages or so (but I think it can be shortened quite a bit). It was actually quite fun, Sudoku like.

Math Is Fun Forum

#1 2009-08-13 12:24:11

how to solve this?

#2 2009-08-13 14:05:32

Re: how to solve this?

#3 2009-08-13 15:09:47

Re: how to solve this?

#4 2009-08-13 18:06:10

Re: how to solve this?

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