Le Monde problem on PIN number

Now that the new school year has started, Christian Robert has picked up solving the Le Monde mathematical puzzles using R again, which leads me to solving them without R… Last week-end’s puzzle is:

Alice’s PIN number is made of four different non-zero digits. She remembers it by noting that when she sums all two-digit numbers that can be made out of those four digits, and multiplies this sum by 7, she gets her PIN number back.

If the PIN number is written \overline{abcd}, it is easy to check that the sum computed by Alice is equal to 33*(a+b+c+d). Thus \overline{abcd}=231*(a+b+c+d).

The sum (a+b+c+d) can take values between 10 and 30. Since 231 is divisible by 3, so is \overline{abcd}. Hence (by the rule used to check for divisibility by 3), (a+b+c+d) is divisible by 3. This makes \overline{abcd} divisible by 9, and so (by the rule used to check for divisibility by 9) (a+b+c+d) is also divisible by 9.

The only possible values for (a+b+c+d) are 18 and 27. It turns out that the only value which works is 18. Hence the PIN number is 18 \times 231 = 4158 (and the sum of those four digits is indeed equal to 18).

Easter eggs

In a comment to his post on Algerians’ disinterest for sex during the Ramadan, Arthur Charpentier shows a graph supporting the claim that around Easter, the French care more and more about eggs and less about Easter itself:

Google Trends : Pâques, oeuf

Assuming Google Trends can be trusted so far back in time, the true story is actually different: the French don’t care less about Easter, they just care less about spelling. If you remove the circumflex from Pâques and search for Paques instead, you will see that the interest in Easter remains roughly constant.

Google Trends: Paques, Oeuf, Pâques

(Note that the correct spelling of oeuf is actually œuf with a ligature, but that is not easily accessible on a French keyboard, nor is it supported by Google Trends.)

The same thing happens at Christmas. It might seem that the French prefer presents to Christmas (Noël vs cadeau)

Google Trends: Noël, cadeau

but they simply ignore the umlaut more and more:

Google Trends: Noël, cadeau, Noel