Self-Describing Number

Find all 10-digit numbers with the following property:

  • the first digit shows the number of 0s in the number
  • the second digit shows the number of 1s in the number
  • the third digit shows the number of 2s in the number, and so on

Let the number be ABCDEFGHIJ. The number of all digits is 10:

A + B + C + D + E + F + G + H + I + J = 10

Therefore, the sum of all digits is 10. Then:

0×A + 1×B + 2×C + 3×D + 4×E + 5×F + 6×G + 7×H + 8×I + 9×J = 10

We see that F, G, H, I, J < 2. If H = 1, I = 1, or J = 1, then the number contains at least 7 identical digits, clearly 0s. We find A > 6 and E = F = G = 0. It is easy to see that this does not lead to solutions, and then H = I = J = 0.

If G = 1, we get E = F = 0. There is a 6 in the number, so it must be A. We get 6BCD001000, and easily find the solution 6210001000.

If G = 0, F can be 0 or 1. If F = 1, then there must be a 5 in the number, so it must be A. We get 5BCDE10000. We don’t find any solutions.

If F = 0, then the number has at least five 0s, and therefore A > 4. However, since F = G = H = I = J = 0, the number does not have any digits larger than 4, and we get a contradiction.

Thus, the only solution is 6210001000.

Cheryl’s Birthday

Cheryl’s birthday is one of 10 possible dates:

May 15
May 16
May 19
June 17
June 18

July 14
July 16
August 14
August 15
August 17

Cheryl tells the month to Albert and the day to Bernard.

Albert: I don’t know the birthday, but I know Bernard doesn’t know either.
Bernard: I didn’t know at first, but now I do know.
Albert: Now I also know Cheryl’s birthday.

When is Cheryl’s birthday?

If Albert knows that Bernard doesn’t know when the birthday is, then the birthday can’t be on May 19 or June 18. Also, Albert must know that the birthday can’t be on these dates, so May and June are completely ruled out.

If Bernard can deduce when the birthday is after Albert’s comment, then the birthday can’t be on 14th. The remaining possibilities are July 16, August 15, and August 17.

Finally, if Albert figures out when the birthday is after Bernard’s comment, then the date must be July 16.

Perfect Day

Dear Diary,
This morning, I woke up at 9 o’clock. My first job was to drink a glass of water and brush my teeth. After that, I ate an apple, 2 bananas, and a toast with almond butter for breakfast. Then, I went to the gym for one hour, took a shower, and came back home at eleven. I cooked a few pieces of chicken with sweet potatoes for lunch; ate everything. In the afternoon, I started working on my projects while listening to classical music. In the evening, I went for the 2nd time to the gym, did cardio for half-an-hour, and after that visited my friend’s apartment. We had together one big pizza for dinner and in the end, I fell asleep on his couch.

As you can see, my entire day was perfect, except that my laptop keyboard broke. Can you figure out what’s wrong with it?

All of the numbers in the text are written using digit symbols except for the ones containing “one” (“one”, “eleven”, “first”). This inconsistency implies that the number “1” key on the keyboard is broken.

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