Kvantik’s Problems

Kvant was a very popular science magazine in Soviet Russia. It was targeted to high-school children and I was a subscriber. Recently I discovered that a new magazine appeared in Russia. It is called Kvantik, which means Little Kvant. It is a science magazine for middle-school children. The previous years’ archives are available online in Russian. I looked at 2012, the first publication year, and loved it. Here is the list of the math puzzles that caught my attention.

The first three problems are well known, but I still like them.

Problem 1. There are 6 glasses on the table in a row. The first three are empty, and the last three are filled with water. How can you make it so that the empty and full glasses alternate, if you are allowed to touch only one of the glasses? (You can’t push one glass with another.)

Problem 2. If it is raining at midnight, with what probability will there be sunshine in 144 hours?

Problem 3. How can you fill a cylindrical pan exactly half-full of water?

I like logic puzzles, and the next two seem especially cute. I like the Parrot character who repeats the previous answer: very appropriate.

Problem 4. The Jackal always lies; the Lion always tells the truth. The Parrot repeats the previous answer—unless he is the first to answer, in which case he babbles randomly. The Giraffe replies truthfully, but to the previous question directed to him—his first answer he chooses randomly.
The Wise Hedgehog in the fog stumbled upon the Jackal, the Lion, the Parrot, and the Giraffe, although the fog prevented him from seeing them clearly. He decided to figure out the order in which they were standing. After he asked everyone in order, “Are you the Jackal?” he was only able to figure out where the Giraffe was. After that he asked everyone, “Are you the Giraffe?” in the same order, and figured out where the Jackal was. But he still didn’t have the full picture. He started the next round of questions, asking everyone, “Are you the Parrot?” After the first one answered “Yes”, the Hedgehog understood the order. What is the order?

Problem 5. There are 12 cards with the statements “There is exactly one false statement to the left of me,” “There are exactly two false statements to the left of me.” …, “There are 12 false statements to the left of me.” Pete put the cards in a row from left to right in some order. What is the largest number of statements that might be true?

The next three problems are a mixture of puzzles.

Problem 6. Olga Smirnov has exactly one brother, Mikhail, and one sister, Sveta. How many children are there in the Smirnov family?

Problem 7. Every next digit of number N is strictly greater than the previous one. What is the sum of the digits of 9N?

Problem 8. Nine gnomes stood in the cells of a three-by-three square. The gnomes who were in neighboring cells greeted each other. Then they re-arranged themselves in the square, and greeted each other again. They did this one more time. Prove that there is at least one pair of gnomes who didn’t get a chance to greet each other.

22 Comments

  1. return0:

    1. Pour glass 5 into glass 2
    2. 0%, its midnight
    3. Dip it sideways into a pool of water down to the diameter, then put a lid on it a lift it out. This one I was unsure of and my solution doesn’t feel like a puzzle solution.
    4. Parrot Lion Giraffe Jackal
    The parrot and jackal were easy to figure they were first and last but the lion and giraffe order I was stuck on for a while. The solution came when I realized that if Lion was third then Hedgehog would know he was third and parrot was first, which he didn’t know (because if he knew giraffe was second and was stuck between parrot and lion for third, he would have known lion when he didn’t “parrot” the last answer)
    5. Two truthful cards can’t be right next to each other because it would mean they represent the same number of lies to their left, therefore the truthful cards have to alternate and there are at most 6. The construction 7-1-8-2-9-3-10-4-11-5-12-6 has six truthful cards (1 thru 6) which means 6 is the true maximum.
    6. 3? seems too easy compared to the rest so maybe I overlooked something
    7. Answer is 9, proof of that fact I haven’t really thought of yet, typing this up quickly before class.
    8. Each corner only shakes two neighbors hands, and there are 12 visits by gnomes to these corners because there are four over 3 rounds. By pigeonhole, 3 gnomes had to visit the corner twice or 1 gnome visited it 3 times and another twice. In the first case, if those three other visit were anything but the center, they would have greeted less than 8 other gnomes and the problem solved. If they did visit only the center and the corners, then they never greeted each other and the problem is also solved. In the second case, the 1 gnome who visited it 3 times only greeted 6 other gnomes meaning he didn’t greet at least two. I believe then that there are at least two pairs of gnomes that didn’t get a chance to greet each other, not just one.

  2. Leo B.:

    Problem 6 is inclear. How do we know how many children and/or grandchildren Olga, Mikhail and Sveta have?

  3. Antonio:

    1: I take the 5th glass and…
    2: Zero because…
    3: I tilt it…
    6: Three? It seems too obvious…

  4. K.:

    @return0
    Why does jackal need to be the last? I got it down to (parrot, lion, giraffe, jackal) and (parrot, lion, jackal, giraffe) as possible solutions, and now I’m stuck as I don’t know how to rule out one of these 2.

  5. tanyakh:

    Problem 6 has lost its meaning in translation. In Russian it is clear that Olga is a girl. This a trivial trick question.

    Problem 8 has another solution using checkerboard coloring.

  6. Q:

    Can you post the full answer to Question 4 please? I don’t agree with the above commenters that Parrot comes first – if so, he would have babbled randomly on the first question and been immediately identified; whereas the question states that only the Giraffe has been identified after Question 1, presumably because he’s the only one to – randomly – say Yes.

    This is driving me crazy!

  7. tanyakh:

    Q,

    Babbles randomly means he can say, yes, no, or anything else. So he can’t be identified if he is the first.

  8. John:

    @K: Jackal was the only one answered “Yes” on the 2nd question. When Parrot answered “Yes” to the last question, he was repeating the last answer he heard.

  9. K.:

    @John
    Thanks. I had misinterpreted the problem – I thought the parrot didn’t repeat an answer from a previous round, only from the current one. Makes sense now.

  10. dEmigOd:

    2.While being obviously 6 times 24 hours, it is not clear a geographical location. So do we need to calculate a proportion of all the possible places with rain and all-day Sun at some point in summer out of the Earth’s surface area?
    6.Would you please post a russian version?

  11. Mr Math:

    I like it!

  12. anon:

    6. unknown – there are 3 siblings in the family – but it’s not known if they are adults or children

  13. Alex:

    6 Not known if the siblings are adults or children – and what their surnames are – Olga coul be Mrs Smirnov her brother Mr Petrov and her sister Mrs Tolstoy so they would not be relevant to this answer

  14. css:

    Is it possible to order this magazine online ?
    t2p2012t2p@gmail.com

  15. Jim:

    @tanyakh:
    If the parrot can babble ‘Yes’, then after the first round, the Giraffe would not be located – the parrot could have said ‘Yes’ and the Giraffe ‘No’, in which case the Giraffe is not located. The Parrot must be assured to not produce a meaningful answer if he’s the first, since this allows the giraffe to be located in the first round

  16. John:

    @Jim:
    Based on the fact the Giraffe was located on the first round, the parrot must have babbled “No” and the Giraffe was the only one answered “YES”.

  17. Максим Новиков:

    Thanks!!!

    Number 3: you can do it without a lid: tilt it till a “large diagonal” becomes horizontal and pour the water in till it could the bottom and, simultaneously, starts overflowing from the top.

  18. joep:

    @4 why is Lion-parrot-giraffe-jackal not correct?

  19. Bart:

    4
    Parrot Lion Giraffe Jackal or
    Parrot Lion Jackal Giraffe
    I am able to eliminate the solution to these two combinations. I can’t follow the explanation of return0.

    Parrot Lion Giraffe Jackal: after round 1 the hedgehog knows that the parrot has to be on position 1 or 2.
    Parrot Lion Jackal Giraffe: only after round 2 the hedgehog knows that the parrot has to be on position 1 or 2.

    So as the same information is known already in an earlier round I favour as solution:
    Parrot Lion Giraffe Jackal

    But I can’t figure out why that information advantage would rule out
    Parrot Lion Jackal Giraffe

    Any suggestions anyone ?

  20. Maks Verver:

    Joep:
    The lion would not answer “yes” to the third question (“Are you the parrot?”), so the lion cannot be first.

    Bart:
    The giraffe is the only animal to answer “yes” to the first question. The jackal is the only animal to answer “yes” to the second question. If the giraffe comes last and the parrot comes first, then the parrot would also answer “yes” (repeating the giraffe’s first answer) to the second question, which would identify the lion instead of the jackal after the second round of answers.

    In the riddle it’s not perfectly clear that the parrot will repeat an answer to a previous question if it’s first in line. But it must be so, otherwise the riddle does not have a definite answer. Ideally the riddle would be less ambiguous, but it’s difficult to clarify this detail without spoiling the parrot’s position (since the distinction matters only when the parrot comes first). I thought Ionica’s version was worded a little better.

  21. Bart:

    Parrot Lion Giraffe Jackall.

    i have read the instructions better. When the parrot is on the first position the parrot “parrots” the last answer of the previous round. (Opposed to providing a random answer on each and every round). Explains it all.

  22. Bart:

    @Joep

    The instructions state that in the 3d round the Parrot will provide the answer Yes.
    Your combination is ruled out as a viable solution as in the 3d round the parrot “parrots” the answer No.