44

Grandpa is thinking of because anagrams to and


16

NB This answer was given when the first line of the puzzle read: "So here is a number below 60" Says Grandpa - since then a further stipulation has been added to restrict the number further to 'between 1 and 60'... I note that Grandpa hasn't specified that the number must be: To this end, I propose that Grandpa might be thinking of: Its anagram is then: ...


14

Ok thinking outside the box... The number could be Since


12

The full list of possibilities that each prime can be adjacent to is as follows: The key ones to notice are Now there are only two possibilities for the other number next to $5$, namely $11$ or $41$. So it must be $11$ next to $5$, then next to that must be either $2$ or $53$. So altogether we have I've tried to illustrate this as follows:


11

Oh boy, here I go swiping again: The number is The difference is Second Try I’m going to use


8

I think this time the path is as follows And the rule is as follows


8

Perhaps the integer between 1 and 60 is the anagram of the number - number > 5 I upvoted El-Guest's answer and it may have been sniped !


7

Answer: Reasoning: If we use a calculator:


6

A lower bound for the number of moves is because This can be done as follows:


5

Partial answer: odd $k$ (no-computers) has a winning strategy, namely This wins because If computers are required for even $k$, I'll leave that for someone else. I'm no good at programming. General notes:


5

Clearly, every power of 2 is a number $x$ as desired. That gives 13 possible $x$ right away. Now let $2^k<x<2^{k+1}$, where $k\leq10$, and consider possible values of $n$. We'll seek the opposite of what was asked for, namely all $x$ such that some $\lceil\frac{x}{2^n}\rceil$ is a multiple of 5. The above is for all $k$, but in particular this means ...


5

This is a partial answer. I'll add more as I go. You are a super spy, typing on your QWERTY keyboard, when suddenly you received an urgent call from your boss, Caesar. Find the law that determines which word goes in which list, You'll find a treasure which will surely make you lift your (eye)brow. Luckily, you also have a gadget which will scan the room ...


5

We need for the following reason:


5

The path is as follows And the relationship is


5

The path is as follows And the relationship is


5

I think the relation is Example: And the path is:


5

The path is as follows And the relationship is


5

Answer Reasoning Now we know that 10yrs = 365*10 = 3650 days(not taking leap years) So now the value you have is which equals


4

I get the picture


4



4

The path is follows And the relationship is


4

The answer is Reason


4

Path(Sorry, but I did on my IPad: Relation:


3

Some musings below: The first thing that strikes us as peculiar about rule B is: Aha! Perhaps the rule is about In particular, So, perhaps these sets represent: To find such a bijection, This does account for all the equivalences in the rule. However, Under this (flawed) approach, the answer to question 2 would then be: If a similar approach were to ...


3

(Not an answer but too long for a comment - computer based) I checked all the possible games for $1 \leq k \leq 10000$ by computer. I discovered the following: I have no idea why this is, but in case it is of some use, the following spoiler contains a table of the winning and losing positions for even $k$ between $2$ and $20$. Note that "Winning" means "If ...


2

The probability is: Reasoning: Because:


2

I dont know if this qualifies for anagram, still the difference is too low


2

Sorry, I did not see that someone posted an answer already. The relationship


2

Relation: The path:


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