Impossible Puzzle

The Lucifer - puzzle ( also known among other names ) is a mathematical puzzle in the field of number theory, which was published by the mathematician Hans Freudenthal.

The mystery impressively demonstrated, as already formulated simply and generally appearing conditions the starting point may be to complex mathematical considerations and also provide a precise and unambiguous solution. It is therefore quite common as an exercise in mathematical training or as an intelligent price puzzle.

The Riddle

There are circulating several versions of the puzzle that are identical in content and differ only in the textual context. A popular version of the " Lucifer - puzzle " led to the designation goes something like this:

The solution

The two figures were sought a and b, for both 1 < a, b ​​< 100 Gauss knows the product m = a · b of both numbers, Euler the sum s = a b.

  • Gauss: "I do not know the two numbers. "

Gauss first determines the prime factorization of m. The numbers a and b it can determine immediately if any of the following occurs:

Since Gauss does not know the numbers at this time, neither of the two cases may be present; the prime factorization of m thus provides at least three factors that are all less than 50.

  • Euler: "It was clear to me. "

Euler looks to the sum s, that the two cases mentioned above certainly are not present. This includes the following values ​​for s from:

The only possible values ​​for s Values ​​of the following set S: = { 11, 17, 23, 27, 29, 35, 37, 41, 47, 53}. At most of these Euler can be sure that Gaussian, the solution can not be read immediately from the product.

Since all values ​​are odd in S, for sure already that one of the numbers a and b is even, the other odd. Further, a and b are in each case less than 53

  • Gauss: "Now I know the two numbers. "

Gaussian can be decomposed product in several ways, of which but gives only a also a sum in S. Among all the possible cases, the following special cases should be noted:

  • Euler: "Then I know them now. "

Euler sees that its sum can be broken down only in one way, which returns one of the above cases. The values ​​of s are eliminated, since Gauss could not determine a unique solution in these cases:

Thus, the value remains 17 Is there actually one (and only one ) Resolution of 17 Gauss can clearly identify as a solution? To this end, all possible decompositions must be checked:

There remains therefore a = 13 and b = 4, a solution that meets the above special case 1. This is actually the only solution that satisfies all conditions.

One way to understand the solution

Those who have trouble Understand the purely mathematical solution should enable even the people Gauss and Euler. After all, everyone got his number: Gaussian, the product 52 and the Euler sum 17 ( This is not the path that brings about the solution to the puzzle, as stated above, but is to make the solution in the context of the dialogue understandable Man. requires only a bit of mathematics [ grade 6 about ] and a bit of logic of language. )

So:

Gauss gives the product 52; Euler the sum of 17

First, Gauss decomposed the number 52 in the possible factors couples:

52 = 4 x 52 = 13 and 2 x 26

Since there is only these may be two pairs of numbers for the product, Gaussian here is the solution actually quite close. He could be the result but only guess, which of course does not come for a mathematician in question. But he also sees that Euler may have indeed received only totals 17 (4 13) or 28 (2 26). Gauss made ​​at the two tables below:

Table 1 represents the sum of 17:

Table 2 is the sum of 28:

The tables show that neither he nor Euler can ( Gaussian ) seen the searched pair of numbers clearly; So he decides to tell Euler that he can not determine the numbers. Previously Euler has of course made ​​his thoughts. Since he has received the number 17 as the sum of the two numbers to guess, he also made ​​to Table 1. He sees that Gauss can only make products that can be formed from at least two factors pairs.

So now it comes to the above dialogue:

Gauss: "I do not know the two numbers. "

(Comment: Since rates are not out of the question, the mathematician is only this simple confession. )

Euler: "It was clear to me. "

( Comment: This clear answer brings Gauss suddenly brooding Why do Euler so well that he can not determine the numbers ( Gaussian ) At this moment Gauss must assume that not 28 may be the sum of the Euler has received since. ? he must rule differently else, for example: " Oh, would have to be " [ namely, the numbers 5 and 23 or 11 and 17 ], but as he says, that it was clear to him, he reveals that his total, can be broken down only in factors that lead to products that have multiple possible factors pairs but this is only the case for the sum of 17 This means:. .. , it must be the numbers 4 and 13 )

Gauss: "Now I know the two numbers. "

(Comment: This response in turn brings Euler brooding How can Gaussian know the numbers because of its utterance Gaussian the solution must therefore already have been very close He considered again the Table 1, where the line is of course immediately on that only. ?. . two factors pairs contains If this would be the solution now, Gauss would have surely the product of 52 broken down into these factors and disassembled the possible summands in turn, possible factors or simply said. . Gauss had table 1 and Table 2 made ​​for this case, may Euler Gauss ' understand thoughts Gauss [ as we indeed know ] have had only two options a, which indeed includes several possible factors couples and one that also contains prime factors pairs [ and thus trivial detachable ] Euler closes but by his statement: .. . " that was clear to me " this sum with the trivial solutions [ linguistically seen ] from So the only other remains. It must be 4 and 13, the numbers).

Euler: "Then I know them now. "

Weblink

  • Number Puzzles There is a more difficult version of this enigma of Robert Sontheimer
  • Easily understood technical programming solution

References

  • Number Theory
  • Mystery
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