# Challenge Riddles and Puzzles: Extend the following to a valid equation

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#### SSequence

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.

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#### fresh_42

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2018 Award
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.

#### fresh_42

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2018 Award
Natural numbers?
Yes, and with a decimal length less than ten.

#### DavidSnider

Gold Member
10^9 - 9^9 = 612579511

#### fresh_42

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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

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2018 Award
15. What is the smallest prime which includes all ten digits exactly once?

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#### 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".

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

#### fresh_42

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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".

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!
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

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2018 Award
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

Gold Member
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?
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

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2018 Award
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

Staff Emeritus
Homework Helper
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2018 Award
Now pour some water into the wine
This must be against some law, but ok ...

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

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2018 Award
This must be against some law, but ok ...

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

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2018 Award
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

Gold Member
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?

#### fresh_42

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

#### DavidSnider

Gold Member
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

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2018 Award
For those who don't see it:

#### fresh_42

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2018 Award
19. Construct $21$ with symbols from $\{\,1,5,6,7,*,/,+,-,(,)\,\}$ but use the digits at most once.

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Homework Helper
15+6

#### fresh_42

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2018 Award
Nice. I should have said: all digits exactly once, or not combined as two digit numbers.

#### BvU

Homework Helper
Moving the goal posts eh ... ?

"Riddles and Puzzles: Extend the following to a valid equation"

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