What’s the difference
- Four digits are added and the same four are multiplied.
- I bet the multiplication gave a much bigger result!
- No, that is what was strange, the result was the same.
- What were the four digits.
- I am afraid I don’t remember.
- Were they four zeroes?
- No. That I would have remembered.
November 26th, 2009 at 2:47 am
4+2+1+1=8
4*2*1*1=8
November 26th, 2009 at 8:53 am
x, -x, 0, 0 also works (it didn’t say I couldn’t choose 1-3 zeroes)
November 26th, 2009 at 10:23 am
Richard, how did you find the digits? Are there more solutions?
Michael, -x is not a digit.
November 26th, 2009 at 1:24 pm
Sorry, Jan, but I have to ask you to think again.
I know some history of computing. Using binary, addition and subtraction were easy. Multiplication was a little more complex, but still fast. Division was slow. Compare; maybe subtract; shift; repeat.
One trick to speed up division was to convert base two numbers to a special radix, with digits of 1 and -1. 3, normally expressed as 011=2+1, would be expressed as 101=4-2+1. This sped up division a little bit by eliminating the compare step.
This problem is not solvable in this radix; 1100 multiply to 1 but add to 0 (base 10). Still, it’s not impossible to presume a radix with 1, 0, and -1 as digits.
November 26th, 2009 at 2:10 pm
I stand corrected!
November 26th, 2009 at 2:56 pm
Suppose we have a set of 4 positive digits, where the sum = the product. 4,2,1, and 1 is obviously such a set. But is it the only set?
Let’s count 1′s
Case 4: 4 1′s. Sum = 4, product = 1. Fail.
Case 3: 3 1′s. Call the fourth digit x. Sum = x+3, product = x. Fail.
Case 1: 1 1. Consider 2,2,2,1. Sum = 7, product = 8. Sum < product. Increasing any digit increases the product more than the sum. Sum remains 1.
Case 2.2: 1,1,2,x. Sum = x+4. Product = 2x. True if and only if x=4.
Case 2.3: 1,1,x,y; x>2, y>2. Fails for 1,1,3,3; product > sum. Increasing either digit increases the product more than the sum. Sum remains < product. Fail.
The solution 1,1,2,4 is unique.
November 27th, 2009 at 10:18 am
positive integer
0,1,2,3 : 0*1*2*3<0+1+2+3 == 0<6
non-zero integer
-1,1,2,3 : -1*1*2*3<-1+1+2+3 == -6<7
November 27th, 2009 at 10:21 am
Sorry, that was for less than.