# A ridiculous puzzle: the end

To recapitulate: I think of two numbers greater than 1. I add them together and give them to Anna. I multiply them together and give them to Mark. Anna and Mark are perfect logicians and perfectly truthful. They have a conversation together about whether they know the numbers I thought of or not.

In Round 1, Mark spoke. He said whether or not he knew what numbers I had thought of (or, equivalently, what Anna’s total was). In Round 2, Anna spoke. In Round 3, Mark spoke. In Round 4, Anna spoke. In Round 5, Marl spoke. In Round 6, Anna spoke. Here is the chart showing what my numbers could be and what Mark’s and Anna’s answer would be in each case. M means that Mark says ‘I know’, A means that Anna says ‘I know’. A dot indicates a ‘don’t know’.

2 3 4 5 6 7 8 9 10 11
2 MA MA M.. MA ..MA MA ….MA ….. ….. MA
3 MA M. .AM MA ….. M. ….. M. ….. M.
4 M. .AM …AM ….. …..A ….. ….. ….. ….. …..
5 MA MA ….. M. ….. M. ….. ….. ….. M.
6 ..MA ….. …..A ….. ….. ….. ….. ….. ….. …..
7 MA M. ….. M. ….. M. ….. ….. ….. M.
8 ….MA ….. ….. ….. ….. ….. ….. ….. ….. …..
9 ….. M. ….. ….. ….. ….. ….. ….. ….. …..
10 ….. ….. ….. ….. ….. ….. ….. ….. ….. …..
11 MA M. ….. M. ….. M. ….. ….. ….. M.

Now it is Mark’s turn again.

Considering the numbers Mark doesn’t yet know the answer to, the only interesting case is if Mark sees 24. 24 can be 2×12 or 3×8 or 4×6.

1. If the numbers are 2 and 12 then Anna will never have said ‘I know’. (This is not visible in the table, which would have to be drawn bigger to show it).
2. If the numbers are 3 and 8 then Anna will never have said ‘I know’.
3. If the numbers are 4 and 6 then Anna will just have said ‘I know’.

If Anna has just said ‘I know’, then Mark can deduce that the numbers are 4 and 6. But if Anna has just said ‘I don’t know’, then Mark can deduce nothing at all, because the numbers could be 2 and 12 or 3 and 8.

Mark can deduce nothing new from Anna’s saying ‘I don’t know’, so the game is at an end.

# A ridiculous puzzle: Round 6

To recapitulate: I think of two numbers greater than 1. I add them together and give them to Anna. I multiply them together and give them to Mark. Anna and Mark are perfect logicians and perfectly truthful. They have a conversation together about whether they know the numbers I thought of or not.

In Round 1, Mark spoke. He said whether or not he knew what numbers I had thought of (or, equivalently, what Anna’s total was). In Round 2, Anna spoke. In Round 3, Mark spoke. In Round 4, Anna spoke. In Round 5, Marl spoke. Here is the chart showing what my numbers could be and what Mark’s and Anna’s answer would be in each case. M means that Mark says ‘I know’, A means that Anna says ‘I know’. A dot indicates a ‘don’t know’.

2 3 4 5 6 7 8 9 10 11
2 MA MA M.. MA ..MA MA ….M ….. ….. MA
3 MA M. .AM MA ….. M. ….. M. ….. M.
4 M. .AM …AM ….. ….. ….. ….. ….. ….. …..
5 MA MA ….. M. ….. M. ….. ….. ….. M.
6 ..MA ….. ….. ….. ….. ….. ….. ….. ….. …..
7 MA M. ….. M. ….. M. ….. ….. ….. M.
8 ….M ….. ….. ….. ….. ….. ….. ….. ….. …..
9 ….. M. ….. ….. ….. ….. ….. ….. ….. …..
10 ….. ….. ….. ….. ….. ….. ….. ….. ….. …..
11 MA M. ….. M. ….. M. ….. ….. ….. M.

Now it is Anna’s turn again.

Considering the numbers Anna doesn’t yet know the answer to, the only interesting case is if Anna sees 10. 10 can be 2+8 or 3+7 or 4+6 or 5+5.

1. If the numbers are 2 and 8 then Mark will just have said ‘I know’.
2. If the numbers are 3 and 7 then Mark will have said ‘I know’ a long time ago.
3. If the numbers are 4 and 6 then Mark will never have said ‘I know’.
4. If the numbers are 5 and 5 then Mark will have said ‘I know’ a long time ago.

In both cases 1 and 3, Anna can now deduce the answer (in cases 2 and 4, she still can’t). Here is the new chart:

2 3 4 5 6 7 8 9 10 11
2 MA MA M.. MA ..MA MA ….MA ….. ….. MA
3 MA M. .AM MA ….. M. ….. M. ….. M.
4 M. .AM …AM ….. …..A ….. ….. ….. ….. …..
5 MA MA ….. M. ….. M. ….. ….. ….. M.
6 ..MA ….. …..A ….. ….. ….. ….. ….. ….. …..
7 MA M. ….. M. ….. M. ….. ….. ….. M.
8 ….MA ….. ….. ….. ….. ….. ….. ….. ….. …..
9 ….. M. ….. ….. ….. ….. ….. ….. ….. …..
10 ….. ….. ….. ….. ….. ….. ….. ….. ….. …..
11 MA M. ….. M. ….. M. ….. ….. ….. M.

So we have the fifth interesting answer in the game.

Mark: Don’t know; Anna: I don’t know; Mark: I don’t know; Anna: I don’t know; Mark: I don’t know; Anna: I know – can only mean that the numbers are 4 and 6. Mark’s number is 24 and Anna’s number is 10.

Next, it is Mark’s turn. Now that he has heard what Ann has said, we shall see what he can deduce.

# A ridiculous puzzle: Round 5

To recapitulate: I think of two numbers greater than 1. I add them together and give them to Anna. I multiply them together and give them to Mark. Anna and Mark are perfect logicians and perfectly truthful. They have a conversation together about whether they know the numbers I thought of or not.

In Round 1, Mark spoke. He said whether or not he knew what numbers I had thought of (or, equivalently, what Anna’s total was). In Round 2, Anna spoke. In Round 3, Mark spoke. In Round 4, Anna spoke. Here is the chart showing what my numbers could be and what Mark’s and Anna’s answer would be in each case. M means that Mark says ‘I know’, A means that Anna says ‘I know’. A dot indicates a ‘don’t know’.

2 3 4 5 6 7 8 9 10 11
2 MA MA M.. MA ..MA MA …. …. …. MA
3 MA M. .AM MA …. M. …. M. …. M.
4 M. .AM …A …. …. …. …. …. …. ….
5 MA MA …. M. …. M. …. …. …. M.
6 ..MA …. …. …. …. …. …. …. …. ….
7 MA M. …. M. …. M. …. …. …. M.
8 …. …. …. …. …. …. …. …. …. ….
9 …. M. …. …. …. …. …. …. …. ….
10 …. …. …. …. …. …. …. …. …. ….
11 MA M. …. M. …. M. …. …. …. M.

Now it is Mark’s turn again.

Considering the numbers Mark doesn’t yet know the answer to, the only interesting case is if Mark sees 16. 16 can be 2×8 or 4×4.

1. If the numbers are 2 and 8 then Anna will never have said ‘I know’.
2. If the numbers are 4 and 4 then Anna will just have said ‘I know’.

So in both cases 1 and 2, Mark can now deduce the answer. Here is the new chart:

2 3 4 5 6 7 8 9 10 11
2 MA MA M.. MA ..MA MA ….M ….. ….. MA
3 MA M. .AM MA ….. M. ….. M. ….. M.
4 M. .AM …AM ….. ….. ….. ….. ….. ….. …..
5 MA MA ….. M. ….. M. ….. ….. ….. M.
6 ..MA ….. ….. ….. ….. ….. ….. ….. ….. …..
7 MA M. ….. M. ….. M. ….. ….. ….. M.
8 ….M ….. ….. ….. ….. ….. ….. ….. ….. …..
9 ….. M. ….. ….. ….. ….. ….. ….. ….. …..
10 ….. ….. ….. ….. ….. ….. ….. ….. ….. …..
11 MA M. ….. M. ….. M. ….. ….. ….. M.

So we have the fourth interesting answer in the game.

Mark: Don’t know; Anna: I don’t know; Mark: I don’t know; Anna: I don’t know; Mark: I know – can only mean that the numbers are 2 and 8. Mark’s number is 16 and Anna’s number is 10.

Next, it is Anna’s turn. Now that she has heard what Mark has said, we shall see what she can deduce.

# A ridiculous puzzle: Round 4

To recapitulate: I think of two numbers greater than 1. I add them together and give them to Anna. I multiply them together and give them to Mark. Anna and Mark are perfect logicians and perfectly truthful. They have a conversation together about whether they know the numbers I thought of or not.

In Round 1, Mark spoke. He said whether or not he knew what numbers I had thought of (or, equivalently, what Anna’s total was). In Round 2, Anna spoke. In Round 3, Mark spoke. Here is the chart showing what my numbers could be and what Mark’s and Anna’s answer would be in each case. M means that Mark says ‘I know’, A means that Anna says ‘I know’. A dot indicates a ‘don’t know’.

2 3 4 5 6 7 8 9 10 11
2 MA MA M.. MA ..M MA MA
3 MA M. .AM MA M. M. M.
4 M. .AM
5 MA MA M. M. M.
6 ..M
7 MA M. M. .. M. .. .. .. M.
8
9 M.
10
11 MA M. M. M. M.

Now it is Anna’s turn again. Anna knows more than she did last time, because she knows whether or not Mark had managed to deduce the numbers, taking into account what he saw in front of him and what she had said before.

Considering the numbers Anna doesn’t yet know the answer to, the only interesting case is if Anna sees 8. 8 can be 2+6 or 3+5 or 4+4.

1. If the numbers are 2 and 6 then Mark will just have said ‘I know’.
2. If the numbers are 3 and 5 then Mark will have said ‘I know’ a long time ago (and Anna will have said ‘I know’ as well).
3. If the numbers are 4 and 4 then Mark will not have said ‘I know’ at all.

So in both cases 1 and 3, Anna will now have deduced the answer (and in case 2 she has known it for some time). Here is the new chart:

2 3 4 5 6 7 8 9 10 11
2 MA MA M.. MA ..MA MA …. …. …. MA
3 MA M. .AM MA …. M …. M. …. M.
4 M. .AM …A …. …. …. …. …. …. ….
5 MA MA …. M. …. M. …. …. …. M.
6 ..MA …. …. …. …. …. …. …. …. ….
7 MA M …. M. …. M. …. …. …. M.
8 …. …. …. …. …. …. …. …. …. ….
9 …. M. …. …. …. …. …. …. …. ….
10 …. …. …. …. …. …. …. …. …. ….
11 MA M. …. M. …. M. …. …. …. M.

So we have the third interesting answer in the game.

Mark: Don’t know; Anna: I don’t know; Mark: I don’t know; Anna: I know – can only mean that the numbers are 4 and 4. Mark’s number is 16 and Anna’s number is 8.

Next, it is Mark’s turn. Now that he has heard what Anna has said, we shall see what he can deduce.

# A ridiculous puzzle: Round 3

To recapitulate: I think of two numbers greater than 1. I add them together and give them to Anna. I multiply them together and give them to Mark. Anna and Mark are perfect logicians and perfectly truthful. They have a conversation together about whether they know the numbers I thought of or not.

In Round 1, Mark spoke. He said whether or not he knew what numbers I had thought of (or, equivalently, what Anna’s total was). In Round 2, Anna spoke. Here is the chart showing what my numbers could be and what Mark’s and Anna’s answer would be in each case. M means that Mark says ‘I know’, A means that Anna says ‘I know’. A dot indicates a ‘don’t know’.

2 3 4 5 6 7 8 9 10 11
2 MA MA M. MA .. MA .. .. .. MA
3 MA M. .A MA .. M .. M .. M.
4 M. .A .. .. .. .. .. .. .. ..
5 MA MA .. M. .. M. .. .. .. M.
6 .. .. .. .. .. .. .. .. .. ..
7 MA M .. M. .. M. .. .. .. M.
8 .. .. .. .. .. .. .. .. .. ..
9 .. M. .. .. .. .. .. .. .. ..
10 .. .. .. .. .. .. .. .. .. ..
11 MA M. .. M. .. M. .. .. .. M.

Now it is Mark’s turn again. Mark knows more than he did last time, because he knows whether or not Anna had managed to deduce the numbers, taking into account what she saw in front of her and what Mark said before.

Considering the numbers Mark doesn’t yet know the answer to, the only interesting case is if Mark sees 12. 12 can be 2×6 or 3×4. If the numbers are 3 and 4 then Anna has just said ‘I know’ in Round 2. If the numbers are 2 and 6 then Anna has just said ‘I don’t know’ in Round 2. In both cases, Mark can now deduce what the numbers are, and say ‘I know’.

So here is the new chart:

2 3 4 5 6 7 8 9 10 11
2 MA MA M.. MA ..M MA MA
3 MA M. .AM MA M M. M.
4 M. .AM
5 MA MA M. M. M.
6 ..M
7 MA M M. .. M. .. .. .. M.
8
9 M.
10
11 MA M. M. M. M.

So we have the second interesting answer in the game.

Mark: Don’t know; Anna: I don’t know; Mark: I know – can only mean that the numbers are 2 and 6. Mark’s number is 12 and Anna’s number is 8.

Next, it is Anna’s turn. Now that she has heard what Mark has said, we shall see what she can deduce.

# A ridiculous puzzle: Round 2

To recapitulate: I think of two numbers greater than 1. I add them together and give them to Anna. I multiply them together and give them to Mark. Anna and Mark are perfect logicians and perfectly truthful. They have a conversation together about whether they know the numbers I thought of or not.

In Round 1, Mark spoke. He said whether or not he knew what numbers I had thought of (or, equivalently, what Anna’s total was). Here is the chart showing what my numbers could be and what Mark’s answer would be in each case:

2 3 4 5 6 7 8 9 10 11
2 M M M M . M . . . M
3 M M . M . M . M . M
4 M . . . . . . . . .
5 M M . M . M . . . M
6 . . . . . . . . . .
7 M M . M . M . . . M
8 . . . . . . . . . .
9 . M . . . . . . . .
10 . . . . . . . . . .
11 M M . M . M . . . M

Now it is Anna’s turn. Anna sees her number, which is the sum of the numbers I thought of, and she has also heard Mark’s statement – ‘I know what the numbers are’ or ‘I don’t know what the numbers are’. Here are some possibilities:

1. Anna sees 7, and hears Mark saying ‘I know’. 7 could be 2+5 or 3+4. From the chart, if the numbers were 2 and 5, Mark would have said ‘I know’ and if they were 3 and 4 Mark would have said ‘I don’t know’. So now Anna knows that the numbers must be 2 and 5.
2. Anna sees 7, and hears Mark saying ‘I don’t know’. 7 could be 2+5 or 3+4. From the chart, if the numbers were 2 and 5, Mark would have said ‘I know’ and if they were 3 and 4 Mark would have said ‘I don’t know’. So now Anna knows that the numbers must be 3 and 4.
3. Anna sees 8, and hears Mark saying ‘I know’. 8 could be 2+6 or or 3+5 or 4+4. From the chart, if the numbers were 3 and 5, Mark would have said ‘I know’ and if they were 2 and 6 or 4 and 4 then Mark would have said ‘I don’t know’. So now Anna knows that the numbers must be 3 and 5.
4. Anna sees 8, and hears Mark saying ‘I don’t know’. 8 could be 2+6 or or 3+5 or 4+4. From the chart, if the numbers were 3 and 5, Mark would have said ‘I know’ and if they were 2 and 6 or 4 and 4 then Mark would have said ‘I don’t know’. So now Anna knows that the numbers must be 2 and 6 or 4 and 4, but she doesn’t know which. Anna doesn’t know the numbers, and never will.

Applying this reasoning throughout, here is the chart after round 2. The first character is M if Mark said ‘I know’ and a dot if he didn’t. The second character is A if Anna said ‘I know’ and a dot if she didn’t.

2 3 4 5 6 7 8 9 10 11
2 MA MA M. MA .. MA .. .. .. MA
3 MA M. .A MA .. M .. M .. M.
4 M. .A .. .. .. .. .. .. .. ..
5 MA MA .. M. .. M. .. .. .. M.
6 .. .. .. .. .. .. .. .. .. ..
7 MA M .. M. .. M. .. .. .. M.
8 .. .. .. .. .. .. .. .. .. ..
9 .. M. .. .. .. .. .. .. .. ..
10 .. .. .. .. .. .. .. .. .. ..
11 MA M. .. M. .. M. .. .. .. M.

So we have the first interesting answer in the game.

Mark: Don’t know; Anna: I know – can only mean that the numbers are 3 and 4. Mark’s number is 12 and Anna’s number is 7.

Next, it is Mark’s turn. Now that he has heard what Anna has said, we shall see what he can deduce.

# A ridiculous puzzle: Round 1

To recapitulate: I think of two numbers greater than 1. I add them together and give them to Anna. I multiply them together and give them to Mark. Anna and Mark are perfect logicians and perfectly truthful. They have a conversation together about whether they know the numbers I thought of or not.

In Round 1, it is Mark’s turn to speak. Here is a chart of what he will say, depending on the two numbers I thought of:

2 3 4 5 6 7 8 9 10 11
2 M M M M . M . . . M
3 M M . M . M . M . M
4 M . . . . . . . . .
5 M M . M . M . . . M
6 . . . . . . . . . .
7 M M . M . M . . . M
8 . . . . . . . . . .
9 . M . . . . . . . .
10 . . . . . . . . . .
11 M M . M . M . . . M

M indicates that Mark has been able to deduce the two numbers; a dot means that he hasn’t.

Mark can deduce the two numbers in the following circumstances:

1. Both the numbers are primes. For instance, if Mark is given 3×5=15, 15 can only be written as 3×5 or 5×3, so Mark will know that the numbers are 3 and 5. I have marked these cases in black in the table.
2. One number is a prime and the other number is its square. For instance, if Mark is given 2×4=8, 8 can only be written as 2×4 or 4×2, so Mark will know that the numbers are 2 and 4. The same thing does not apply to cubes. If Mark is given 2×8=16, that can be 2×8 or 4×4, and Mark can’t tell which is is. I have marked these cases in red in the table.

In every other case, Mark can’t tell what the numbers are. For instance, if he is given 3×4=12, he will see 12 but he will not be able to decide whether that comes from 3×4 or 2×6.

Next, it is Anna’s turn, and we shall see what she can deduce.

# A ridiculous puzzle

I think of two numbers greater than 1 (they may or may not be the same number).

I add these two numbers together and give them to my friend Anna. I multiply the same two numbers together and give them to my friend Mark. Both my friends are perfectly truthful and perfect logicians.

1. M says to A: ‘I don’t know what your number is’.
2. A says to M: ‘I don’t know what your number is’.
3. M says to A: ‘I don’t know what your number is’.
4. A says to M: ‘I don’t know what your number is’.
5. M says to A: ‘I don’t know what your number is’.
6. A says to M: ‘I know what your number is’.
7. M says to A: ‘I know what your number is’.

Question. What are Anna’s and Mark’s numbers? (Or, equivalently: what numbers did I originally think of?)

# More self-indulgence

You can tell I’ve got some serious work on when I’m on here instead. Here’s another Proof from THE BOOK.

Write down 10 numbers. Any 10 whole numbers. They don’t even all need to be different.

Then:

1. either one of them is a multiple of 10,
2. or a sequence of them adds up to a multiple of 10.

For instance, if I write down 3 1 4 1 5 9 2 6 5 3, then 4+1+5=10 and 5+9+2+6+5+3=30.

(Note: the division into case 1 and case 2 is purely artificial. A mathematician would be happy with “a sequence of 1 number adds up to a multiple of 10” but real people aren’t.)

# Pure self-indulgence

Well, almost pure.

I was leafing through Aigner and Ziegler’s Proofs from THE BOOK, which is a compendium of the most beautiful results in mathematics, and I came across a very simple puzzle which Paul Erdös used to use when he wanted to see if someone was really a mathematician. Here it is, without the algebra:

Think of the numbers from 1 to 100. Prove that if you make a collection of any 51 numbers between 1 and 100, at least one number in your collection will be divisible by some other number in your collection.

I am not a real mathematician. I am not patient enough. Knowing I’d regret it for the rest of my life, I read the solution. It is indeed very beautiful. To atone for my crime and assuage my anguish, here is the proof in a form that I think everyone will be able to follow. Try it. It’s fun.