Friday Puzzler - 3388

Hint

Alex Y

Exactly one set of parentheses is required to force the calculation order.

Re: Hint

Larry Barrett

No one seems to have any ideas on this one. I don't either. But I offer this up as a little humor while waiting for spring to really arrive here in MD.

Observe that 8-3+8 = 13 and 8+3 = 11, so (8-3+8) + (8+3) = 24. The only problem with this solution is that it requires an extra 8.

Unless it is written this way: 8+(-3 [+8) +3] and you apply the rarely used inverse Polish notation rules.

Re: Hint

Alex Y

You have made one assumption, which is valid: The four digits represent 4 numbers (i.e., 33, etc., are not part of the solution.)

The minus sign can be used only for subtraction, not for negation.

I don't follow your conventionfor showing reverse polish calculation. But since you mentioned RPN, I will tell you that on my HP calculator, I get the stack four deep. I.e., I press enter after putting in each of the first three numbers, and after keying in the fourth number, hit three operators to get the answer. I haven't thought through whether this is giving away too much or not much at all. It certainly makes brute force a better option -- only 384 possibilities to check out, if I am correct.

Remember the first hint-- the one pair of parentheses is REQUIRED (when using "regular" infix notation) to make sure the calculation is done in the correct order. That is not the case with the example partial attempts you showed.

Re: Hint

Larry Barrett

I called my notation the inverse Polish notation (IPN) to distinguish it from the legitimate (but somewhat obscure) Reverse Polish Notation (RPN). I do not have a real solution using any legitimate notation, so far.

Hint 2

Alex Y

This is a big one.

The answer is unique.

That might not sound like a big hint, but thinking of what it implies...

Hints

Alex Y

The impact of the previous hints is not immediately obvious.

Hint 1 - Exactly one set of parentheses is required to force the calculation order.

Normally, all multiplications and divisions are performed first, left to right, then all additions and subtractions. left to right. A set of parentheses breaks that order. For instance, if our goal was 25 rather than 24, we could write 3*(3+8)-8. The parentheses are required to have the 3+8 performed first.

Hint 2 - The answer is unique.

Both addition and multiplication are commutative. So if you have an expression that contains A*B or A+B, you could swap A and B and get the same result with an expression that is different (unless A and B are the same). For instance, the formula above for 25 is not unique -- you could just as well write (8+3)*3-8

Final Hint

Alex Y

An 8 is outside the parentheses and the other three numbers are inside.

Re: Final Hint - in spite of all your hints,

Larry Barrett

I am still stumped.

If the solution is unique, and if the 8 precedes the parens, then it must be followed by a divide sign and the value inside the parens must equal 1/3 so that 8/(1/3)=24.

Or, if the 8 follow the parens, then it must be preceded by either a minus or divide sign. If a minus sign, then the value inside the parens must equal 32 so that (32)-8=24; if a divide sign then the value inside the parens must equal 24x8 =192 so that (192)/8=24.

The best I have been able to do is come up with expressions that =25 or 23. I suppose that my average solution on alternate days =24, so I claim victory.

So close!

Alex Y

From the hints, you have correctly determined that the answer is either 8/(1/3) or (192)/8.

(32)-8 is not an option, since the parentheses are not required to make the subtraction come last.

The last operation performed in the parentheses is a subtraction, since an addition would not be unique (switch the terms) and a multiplication or division would not require the parentheses to assure correct operation order (numbers could be rearranged to achieve the same result).

Re: So close!

Larry Barrett

Oh, oh, call on me!

How about 8/(3-8/3). Why was this so hard?

Correct, but I don't know

Alex Y

Why it was so hard. I couldn't get it either. Maybe we just aren't "wired" to use division to get larger numbers.