Puzzles - Need help !

[mashajamo]

Three men order a bottle of wine at $30 to drink with their meal. They each pay $10 for it. The waitress asks the manager who only has a bottle at $25, so he gives the waitress the wine and $5 change. She returns to the table but gives only $3 back keeping $2 for herself. Since each man paid $10 and received $1 change he has paid $9. But 3 X 9 = $27 which with the $2 kept by the waitress makes $29. Where did the $1 go?


You are a prisoner with two guards, each guarding one door to your cell. One guard always lies and one is always truthful, but you do not know which is which. You may leave your cell by one of the two doors. One leads to the execution block and death. The other to freedom. You may ask only one question, which you may address to either of the guards but not to both. What do you do?

Certain numbers can be expressed as a function of their own digits, while maintaining the order of the digits. For example 36=3! x 6. Can you find the formulation for 71, 119, 720, 733 and 936?

 

[Hamilton]

hum

the first is just plain silly so someone else can answer that .ok then 25 + 2 =27 = 3 x 9

you ask either guard "what would the other guard answer if i asked which door is freedom ? "

if you are talking to the guard who always tells the truth he knows the other guard is a liar and so he will point you to the execution door , you take the other door and freedom awaits.

if you are talking to the guard who always lies he knows the other guard always tells the truth but he himself will lie about what the truthful guard will say , so again he will point at the execution door , you take the other door and freedom awaits .
ie the same decision so it dont matter what guard your talking too.

hum functions on digits , so factorial , euler totient im finding it hard to think of anymore common functions i suppose you can square and cube and square root etc.

71 = squareroot ( 7! + 1 ) that looks like it could be the answer to me , not to sure on the rest they are tough

720 = (psi(7))squared x 20 this is a guess cant see it being the answer psi(7) is 6 its the euler totient

wow ive been looking at this for over an hour time to admit defeat

 

[DevonMatthews]

how do you define a function? If your allowed to use a square root/square whats stopping you from doing

119 = [1^0 + 1^1 + 9^(ln(117/ln9))]

This is expressed as 3 unique functions

f1(x) = x^0
f2(x) = x^1
f3(x) = x^(ln117/ln9)

 

[Hamilton]

very nice

i suspect square root is alright ( cause you can do the symbol without a number ) but cube roots etc and your very clever function actually involve other digits i suspect are no-noes.

probably ain't gonna see the euler totient function either , reckon these questions aren't aimed at mathematicians , so its probably just + - * % and factorial .

ive been banging my head against a wall trying to get these (im a very sad individual) so if anyone gets some more please please post your solutions .

mashajamo i presume you havent got solutions?

 

[AntH67]

I think these are the intended solutions (though some have more than 1):

119 = -1 + (-1 + √9!)!

720 = (7 - 2^0)!

733 = 7 + 3! + 3!!

936 = (√9!)^3 + 6!

 

[DevonMatthews]

I think these are the intended solutions (though some have more than 1):

119 = -1 + (-1 + √9!)!

720 = (7 - 2^0)!

733 = 7 + 3! + 3!!

936 = (√9!)^3 + 6!

If you figured figured that out yourself then you sir are a genius.