Riddles and Puzzles: Extend the following to a valid equation

[mfb]

The solution to number 9. I was looking for was the pair $(4,13)$. The clue was to rule out all possibilities of the form $pq$ which allowed a unique factorization by comparison to the according level of information.

What was the error in my reasoning then?
pq does not allow a unique factorization as (1,pq) is a valid pair of numbers. You didn't specify that 1 cannot be a number.

 

[fresh_42]

What was the error in my reasoning then?
pq does not allow a unique factorization as (1,pq) is a valid pair of numbers. You didn't specify that 1 cannot be a number.

Yes, you are right. I had forgotten to mention that the numbers are properly between 1 and 100.

I don't like chess puzzles that don't specify who is who and who moves next.

Well, I did specify it:

• Meier vs. Carlsen (This implies Meier white and Carlsen black.)
• How did Carlsen win? (This implies black to draw.)

 

[fresh_42]

12.

In which order are the cups filled with coffee?

 

[lpetrich]

My solution:

Spoiler

The coffee is poured into the top chamber, and as that chamber is filled, the coffee level reaches the pipe on the left.

When it does so, it pours into the left chamber. As it does, it reaches the left pipe from it. That pipe is blocked at the chamber, so the coffee continues to fill that chamber. Cup 4 thus never receives any coffee. When it gets to the level of the right pipe from that chamber, it flows into that pipe, but it is blocked at the spout end. Cup 9 never receives any coffee, and the coffee continues to fill the left chamber, filling it and the pipe leading into it.

It fills the top chamber until it reaches the pipe on the right, and it then pours into the top right chamber, and then the bottom right one. When it fills the bottom right one, it starts filling the top right one until it meets the pipe on the right. The coffee pours into that pipe but is blocked at the joint. Cup 7 never receives any coffee.

The top right chamber continues to become filled until the coffee level reaches the pipe to the left, and it then pours into that pipe. It is unobstructed, and the coffee pours into Cup 5.

Thus, cup 5 is the only cup of these four that ever receive any coffee.

 

[fresh_42]

No chess fans here?
https://www.physicsforums.com/threa...o-a-valid-equation.970213/page-3#post-6171170
I'm a bit disappointed.

14. How many numbers are there that contain their length as a digit?

 

[SSequence]

No chess fans here?
https://www.physicsforums.com/threa...o-a-valid-equation.970213/page-3#post-6171170
I'm a bit disappointed.

For what it's worth, I have beaten lvl-1 stockfish (on here) a couple of times (without any move-backtracking) ... and losing many times :P.

I actually looked at it for quite some time yesterday. I just couldn't find a way to find a quick victory. Eventually, I searched-up and just looked at the game, but well ... I am still not sure. I thought of that particular move (for black). The response of white that I couldn't resolve isn't taken in actual game (likely for good reasons, but somehow I don't get it). But maybe if I took up pen and paper and tried to sketch every possibility, I could be able to do it.

But since I already watched, it would be better for me not to post anything.

 

[fresh_42]

But since I already watched, it would be better for me not to post anything.

For one there is the thread to run into a position to lose the queen by chess through the knight, and the other option is to run away with the king which ultimately loses a knight or allows the pawn to convert.

 

[SSequence]

I just forgot about pawn promotion completely. It should likely be solveable with promotion.

 

[DavidSnider]

No chess fans here?
https://www.physicsforums.com/threa...o-a-valid-equation.970213/page-3#post-6171170
I'm a bit disappointed.

14. How many numbers are there that contain their length as a digit?

Natural numbers?

 

[fresh_42]

Natural numbers?

Yes, and with a decimal length less than ten.

 

[DavidSnider]

Spoiler: spoiler

10^9 - 9^9 = 612579511

 

[fresh_42]

Spoiler: spoiler

10^9 - 9^9 = 612579511

Correct, but you could have said how you did it. Not because it is difficult, or can be guessed by the formula, but because it is an important technique in counting and often saves time, programming or even an induction:
It is often easier to count the complement!

 

[fresh_42]

15. What is the smallest prime which includes all ten digits exactly once?

 

[pbuk]

I have not seen the word 'cipher' used that way before in English - I think most people would ask "What is the smallest prime which includes all ten decimal digits exactly once".

Spoiler

The word 'digit' probably makes it easier to get to the root of the problem.

 

[fresh_42]

I have not seen the word 'cipher' used that way before in English - I think most people would ask "What is the smallest prime which includes all ten decimal digits exactly once".

Spoiler

The word 'digit' probably makes it easier to get to the root of the problem.

Thanks, corrected. The difficulty is to distinguish between digit as a number and digit as a position, cause you also say: estimate up to three digits, which would be position. As I meant the symbol, and neither value nor position, I thought cipher would be adequate.

 

[pbuk]

15. What is the smallest prime which includes all ten digits exactly once?

I seem to have killed this, so I suppose I had better answer it and we can move on!

Spoiler

There is no such prime. The sum of the digits 0...9 is 45 which is divisible by 9, and so any integer which includes each of these digits exactly once is also divisible by 9.

 

[fresh_42]

16. A man drove his car one mile to the top of a mountain at the rate of fifteen miles per hour. How fast must he drive one mile down the other side in order to average thirty miles per hour for the hole trip of two miles?

 

[Borg]

16. A man drove his car one mile to the top of a mountain at the rate of fifteen miles per hour. How fast must he drive one mile down the other side in order to average thirty miles per hour for the hole trip of two miles?

Spoiler

It's impossible. It takes 4 minutes to go 2 miles @ 30 MPH but the first half of the trip already took 4 minutes.

 

[fresh_42]

17. You have a glass of 200 ml of water and one with 200 ml of wine. Now pour some water into the wine and after mixing, pour back so much again, until the water glass has again 200ml. 90% are still water. How big is the percentage of wine in the wine glass?

 

[Orodruin]

Now pour some water into the wine

This must be against some law, but ok ...

Spoiler

The water glass now contains 90% (180 ml) of water and 10% (20 ml) of wine. Thus, the wine glass must contain the remainder, i.e., 180 ml of wine (90%) and 20 ml of water (10%).

 

[fresh_42]

This must be against some law, but ok ...

Spoiler

The water glass now contains 90% (180 ml) of water and 10% (20 ml) of wine. Thus, the wine glass must contain the remainder, i.e., 180 ml of wine (90%) and 20 ml of water (10%).

It is semi illegal.$^*)$

$^*)$ Only allowed for the female half of the population.

 

[fresh_42]

This time I will leave the usual story about sons and heritages etc. but expect a little use of mspaint or similar:

18. Can this field be partitioned in four equally sized, congruent parts?

 

[DavidSnider]

This time I will leave the usual story about sons and heritages etc. but expect a little use of mspaint or similar:

View attachment 242868

18. Can this field be partitioned in four equally sized, congruent parts?

Spoiler: spoiler

 

[fresh_42]

Spoiler: spoiler

View attachment 242893

I can see the solution shine through, but hell, what have you done? Did you include a proof of congruence?

 

[DavidSnider]

I can see the solution shine through, but hell, what have you done? Did you include a proof of congruence?

No I was just playing around with GeoGebra. Wouldn't know how to show the proof.

 

[fresh_42]

For those who don't see it:

 

[fresh_42]

19. Construct $21$ with symbols from $\{\,1,5,6,7,*,/,+,-,(,)\,\}$ but use the digits at most once.

 

[BvU]

Spoiler

15+6

 

[fresh_42]

Spoiler

15+6

Nice. I should have said: all digits exactly once, or not combined as two digit numbers.

 

[BvU]

Moving the goal posts eh ... ?

 

[fresh_42]

Moving the goal posts eh ... ?

I know, my bad. But 15+6 is too easy.

 

[DavidSnider]

19. Construct $21$ with symbols from $\{\,1,5,6,7,*,/,+,-,(,)\,\}$ but use the digits at most once.

Spoiler: spoiler

6 / (1 - (5/7))

 

[fresh_42]

20. The king gives the convicted one last chance to save his life. The prisoner receives 50 white and 50 black balls, which he may arbitrarily distribute to two identical-looking vessels. The next day he has to randomly pick a vessel and draw a ball out of it. He will be executed at black, pardoned for white.

How must the prisoner distribute the balls so that his chances to survive are as high as possible?

 

[pbuk]

15. What is the smallest prime which includes all ten digits exactly once?

There are of course some 'cheating' answers e.g. 1,023,457,986₂₃ = 1,808,433,654,721₁₀ which is prime (probably not the smallest such prime though).

Or to take the radix cheat further, how about 2_134567890?

 

[mfb]

How must the prisoner distribute the balls so that his chances to survive are as high as possible?

Spoiler

(1 white) and (49 white 50 black).
Proof that it is optimal: More than 50% in a vessel requires more white than black there which forces less than 50% in the other one. 49/99 is the closest you can get to 50% winning chance if you have fewer white than black balls, and 100% winning chance is clearly optimal for the other vessel. This strategy is optimal if one vessel has fewer than 50% winning chance. This strategy also beats 50% in both, therefore it must be optimal overall.