What is more difficult: to show that something is possible or to show that it is impossible?
the professor Phi asks his Alpha students and Beta :
– Take a four-digit number, turn it over and add these two numbers, then give me your results.
Alpha gives 7.106 and Beta, 3.213. Professor Phi thought for a moment and said:
– One of you was wrong.
Who ?
Responnse :
Beta was wrong in its calculation. To see it, let’s examine the result we get from the number abcd, i.e. 1.000 at + 100 b + 10 vs + d. By adding to it dcba, all calculations done, we get the result R = 1.001 (at + d) + 110 (b + vs). For this number to be equal to 3.213, it is necessary that the units digit of at + d so that is 3 at + d that is to say equal to 3 because 13 would imply R> 13,000 which would make the equality R = 3.213 impossible. We get 110 (b + vs) = 210 which implies that 11 divides 21 … which is false. Beta was therefore wrong in its calculations.
For Alpha, it is possible that his calculations are correct. For example, he could have started from the number 2,734 but this is not the only possibility. Showing an impossibility is often more difficult than showing a possibility.
Another method
The method used by Phi is general but, in this precise case, Phi could also have noticed that 110 and 1001 being both divisible by 11, R must also be so or 3213 is not …
Learn more about Hervé Lehning
Normalian and agrégé in mathematics, Hervé Lehning taught his discipline for more than forty years. Crazy about cryptography, member of the Association of Cipher and Information Security Reservists, he in particular unraveled the secrets of Henry II’s cipher box.
- His blog MATH’MONDE on Futura
- the last book by Hervé Lehning :
Also to discover: The universe of secret codes from Antiquity to the Internet published in 2012 by Ixelles.
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