Lowest Value Function
Define a function F as follows
F ( w , x , y , z ) = 8 1 ( S − T )
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For certain positive integers a , b , c , d and for all w , x , y , z , this function F ( w , x , y , z ) always returns the lowest of the values w , x , y , z . For example
F ( 4 , − 3 , 2 , 1 ) = − 3 .
Let a , b , c , d be integer digits of a 4 digit integer A = a b c d . What is the value of A ?
The answer is 1124.
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3 solutions
@Terrell Bombb I've elaborated on my solution. The first equation is a common computing trick to define the minimum function.
First, observe that a = b . Now, for finding d , we put w = 0 , x = 0 , y = 0 , z = − 1 . Then we equate the function to − 1 , as it is the least number. And hence we get value of d = 4 .
Similarly, c = 2 , b = a = 1 .
Hence, the answer is 1 1 2 4
Many thanks to the author for making me realize that this is possible!
Another way of approaching this problem (in a straight forward way) is to use the fact that
min ( a , b ) = 2 a + b − ∣ a − b ∣ .
So this gives us
min ( a , b , c ) = min ( min ( a , b ) , c ) = 2 2 a + b − ∣ a − b ∣ + c − ∣ 2 a + b − ∣ a − b ∣ − c ∣ .
We then repeat this once more.
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Where can I read more about the idea of your solution? What topic is this sir?
I have no idea how you came up with that solution. And I also don't understand why a=b in the first place.
I calculate the S and T value for (w,x,y,z) in this way:
S(0,0,0,-1) = -d and T(0,0,0,-1) = d
S(0,0,-1,0) = -2c and T(0,0,-1,0) = 2c
S(0,-1,0,0) = -2b-2 (if b > -1) and T(0, -1,0,0) = 2 - 2b (if b > -1)
S(-1,0,0,0) = -2a-2 (if a > -1) and T(-1, 0,0,0) = 2 - 2a (if a > -1)
Than i wright 1/8*(S-T)=-1 and I calculate a, b, c, d a = 1, b = 1, c = 2, d = 4 Eventually i join all togheter A = 1124
I did something similar, using <0,0,0,1> permutations, which only gave a,b. I see here <0,0,0,-1> would have been better ! Thanks.
Let's find this function by simply using these two facts:
1. min ( a , b ) = 2 a + b − ∣ a − b ∣
2. min ( a , b , c ) = min ( min ( a , b ) , c )
Applying it once, we obtain:
min ( a , b , c ) = min ( min ( a , b ) , c ) = 2 2 a + b − ∣ a − b ∣ + c − ∣ 2 a + b − ∣ a − b ∣ − c ∣ . = 4 a + b − ∣ a − b ∣ + 2 c − ∣ a + b − ∣ a − b ∣ − 2 c ∣
We then repeat this once more to obtain:
min ( a , b , c , d ) = min ( min ( a , b , c ) , d ) = 2 4 a + b − ∣ a − b ∣ + 2 c − ∣ a + b − ∣ a − b ∣ − 2 c ∣ + d − ∣ ∣ ∣ 4 a + b − ∣ a − b ∣ + 2 c − ∣ a + b − ∣ a − b ∣ − 2 c ∣ − d ∣ ∣ ∣ = 8 a + b − ∣ a − b ∣ + 2 c − ∣ a + b − ∣ a − b ∣ − 2 c ∣ + 4 d − ∣ a + b − ∣ a − b ∣ + 2 c − ∣ a + b − ∣ a − b ∣ − 2 c ∣ − 4 d ∣ .
Thankfully, this is in the form required by the question, and we see that a b c d = 1 1 2 4 .