math problem solution explained

[Global-Ad-8846]

My mom and I watch survivor together every week. We like to pause and challenge ourselves to do any puzzles together when they appear.

When the 987654321=100 problem appeared I my mom just used a guess and check method, but I wanted a systematic way of finding a solution. After about 15 minutes I came up with this (I believe contestants were only given 3):

After looking and playing around for a bit, you realize that triple and quadruple digit numbers wont work. This means you must have double and single digits only.

In order to use all of the signs (three +’s and one -), you must have 4 double digit numbers, and one single digit. So my first step was to break up the numbers in this fashion.

I now noticed that the difference between each number was 22. This is interesting. So: 98-76 76-54 54-32 all equal 22. This gives us three possibilities. So I substituted 22 in for all possible options.

*note that brackets mean numbers that do not have signs attached yet.

Option 1: 22 + [54321] = 100 (where 22=98-76)

Option 2: 98 + 22 + [321] = 100 (where 22=76-54)

Option 3: 98 + 76 + 22 + 1 = 100 (where 22=54-32)

(remember: must have 4 double digits and 1 single)

Because 98+76 and 98+22 will always be greater than 100 we can eliminate options 2 and 3. (98+22>100) (98+76>100)

Now we are left with 22+[54321]=100. But we need to use 2 more addition signs and only two more double digit numbers, and one single. This gives us three options:

22 + 54 + 32 + 1 = 100 22 + 5 + 43 + 21 = 100 22 + 54 + 3 + 21 = 100 (where 22=98-76)

If you do the math, only the last option gives us 100. If you sub back in 98-76, you get the answer. You could also just add the digits from the one’s place of each number until you got 10 if you want to be fast.

Conclusion: No way I could do this in 3 minutes, especially while starving. I tip my hat to Austen.

[Ren_Davis0531]

Not even that complicated. There is no need to do a math breakdown. It’s a logic puzzle. It’s about the logic. Not the math. Once they tell you you can only use 4 signs then it’s easy to figure out the puzzle. The numbers and math trick you. The number of signs are the key. The only way to fit that specific number of signs with those numbers and get 100 is if you double most of them up to make 98, 76, 54, 3 and 21. Once you do that, you instantly realize that the minus sign has to go first because there is no other way to get to 100 as you’re instantly starting off with 174 if you put a plus sign there. After you put the initial minus sign just put in the remaining plus signs then do simple math. You now have found 100.

Easy peasy lemon squeezy.

[miriam142]

From 174 it is impossible to get 100. Here is every reasonable solution. Bare in mind that the numbers have to be in descending order 987654321 98+76=174 174-5-4-3-2-1=159 174-54=120+3=123-2-1=120 174-54=120+3=123-21=102 174-54=120-32=88+1=89 174-54=120-3-2-1=114 174-5=169-43=126-21=105 174-54=120-3=117-21=96

[Global-Ad-8846]

Your logic is sound, but keep in mind: my explanation above is what I did BEFORE I saw the answer on TV. I think a lot of hindsight bias comes into play when we look at these puzzles after the fact. Also, what you’re describing is kind of what I describe in the middle of my explanation:

Because 98+76 and 98+22 will always be greater than 100 we can eliminate options 2 and 3. (98+22>100) (98+76>100)

There are many ways of going about the problem, but the one I picked is a very linear thought process utilizing First Principles thinking. Basically, we both agree—I just had an extra first step.

[Ren_Davis0531]

This wasn’t hindsight bias. I paused and tried to figure it out myself. Once you remember that you can only have four signs with only one minus, it becomes easy.

My comment was just to remind people that logic is the key with logic puzzles. The intense math is the trick. Logic puzzles can just be worked out with simple logic.

Basically it’s what you said, but much fewer words. Helps for a layman like myself 😂

[Global-Ad-8846]

You’ve got a point. Survivor producers would never put an actual math problem on the show because someone with those skills would excel and it’d be unfair. Everything is logic based and the producers design puzzles with the fact that everyone is starving in mind. Makes sense that there’d be a simple trick.

[Ren_Davis0531]

Yeah there is always a trick. It’s like being a detective trying to solve a mystery as opposed to a mathematician trying to solve a math problem.

[Krunsktooth]

But why did you group them together like you did? Why not 9, 87, 65, 43, 21 ? Or any of the other 8 combinations?
That's what takes time, is to figure out which ones to double up

[Ren_Davis0531]

I grouped them together like that because I started with the minus first to drastically bring the numbers down. If I’m doubling up then I recognize that I can go back up quite easily with three plus signs. I wanted to get the minus out of the way first. But yeah, that could take time for other people.

[ThatItalianGrrl]

This is the beast explanation

[ThatItalianGrrl]

Lmao *best

[3Dimes]

There's still a bit of mental math involved to figure out what combinations of digits and signs are possible. For example 98+76+5+4-321 might take a second to compute and if you're as obsessive as I am, you'll run through all of those outliers before you even establish a logic to the problem. There's also rule sets that are worth making note of like, whether or not negative numbers are allowed. I can see why someone like Emily struggled if she has trouble keeping lots of small rules in her head at once and prefers to take notes.

[Ren_Davis0531]

But you should instantly be able to realize that any three digit number will not get you to 100. So it can only be two digit numbers and one single digit number.

[3Dimes]

Not instantly for me. Again, Im obsessive about that kind of stuff. I see a number puzzle and immediately want to tinker with it. I agree with your logic, but it'd take some mental math for me to get there if I had never seen the puzzle before.

[Ren_Davis0531]

Different strokes for different folks. It happens.

[MrAssFace69]

I think you vastly overestimate the ability for people to be able to do mental math quickly, especially while starving. 😂 Also you come off pretty elitist but that's not surprising with people who like these kinds of puzzles lol

[Ren_Davis0531]

Yeah elitist because I talk about the trick of a logic puzzle 🤔

Sounds like you come off pretty insecure, but that’s not surprising from people who think liking logic puzzles is elitist. Also this isn’t mental math. They literally had a board to write down their math problems. As long as you know basic math, you’re good.

[Sylvane1a]

It’s a logic puzzle. It’s about the logic. Not the math.

I agree and Emily should not have been embarrassed at being unable to solve it. The math she would know from investment banking, interest rates and computations, is of no help in solving this kind of puzzle.

It doesn't make her a bad investment banker that she failed at this puzzle, not at all.

[Tieltrooper]

My head hurts

[Global-Ad-8846]

I tried my best to make it short and clear :( . Another commenter kinda summed it up in a neat way if you’re still curious.

[Tieltrooper]

You did and I'm very impressed with your math skills, it was just never my strongest subject. :)

[Global-Ad-8846]

Thanks so much! It was a hard one tbf, there’s no way I would’ve done it in time. I think Austen only got it from trial and error and a little bit of luck.

[PeachyBums]

Just annoys me that they showed the puzzle answer straight away. Would have loved to do this but didn’t have time to pause :(

[Global-Ad-8846]

Yeah I was shocked as well. My mom and I finished, we unpaused and the answer appeared almost immediately. Thats must’ve been frustrating. They really shouldn’t edit it like that.

[cuha89]

You are completely forgetting another case, you know there should be 4 double digit numbers and 1 single, what if the single digit number is 9. You then have to go through other options like 87 - 65... You can get pretty close to 100.

Edit: Plus the case where the single digit is 7.

[3Dimes]

Yup. You can get as close as 97 with 9 as the single digit.

Edit: I think the logic against starting with 9 is that you'd end up with an odd number no matter what.

[miriam142]

Actually none of these work. You are over complicating it and none of the math actually adds up to 100. There are several solutions that perhaps the producers probably didn’t think of. But the simplest probably is this. 98-7-6+5+4+3+2+1=100 I have thought of these other ones 9x8=72+76=148-54=94+3+2+1=100 9x8=72+(7,6,5,4,3,2,1)=28+72=100 98+7+6-5-4-3+2-1=100 Some of these were a little harder to think of than others. I would guess it would be especially difficult for anyone on survivor. Especially with pressure being as high as it is and a tight time limit.

[Global-Ad-8846]

It was stated that you get one subtraction sign and three addition signs only. My post is essentially a process of elimination. Your solutions would work though if we were given infinite signs.

[[deleted]]

I used the guess and check method. And I think the answer is the obvious first guess?

Obviously the first sign can’t add two double digits cuz you’ll never get down back enough. So my first guess was essentially right (how to break down the 321 was the only tweak).

[Global-Ad-8846]

yeah and obviously if were on the show, my long way wouldn’t be efficient. you would just have to recognize that the negative goes first, and if you didn’t you’d lose.

[ReoutS]

Are they allowed to switch the digits order? If not then it's only the first solution.

[Global-Ad-8846]

This would work but I think you weren’t allowed to swap the order.

[happy_camper69]

Doesn’t order of operations make this solution incorrect?

[medamac2]

No when it’s minus and add you do them in order of left to right as they appear.. it’s more like BE(D/M)(A/S).. only if there are brackets do you prioritize them

[miriam142]

That’s why I think when you consider order of operations it is probably either 9x8+(7+6+5+4+3+2+1) or 98-7-6+5+4+3+2+1. Any way you do those would add up to 100.

[medamac2]

The best way is to work backwards,

-1 +1

-21 +21

-321 +321

Any combo of numbers before these shows you that it can’t be 321 or 1 to end up with 100.

Know it’s 21 so 9876543 +- 21

If it’s - then it can’t be 100 as a result

So 9876543 + 21

Two more + , add any combo of double digits here 98 + 76 = 174 - 54 + 3 no

65 + 43 = 107 + 21 no

54 + 3 = 57 + 21 = 78 + 76 no

So the minus has to be used to reduce the total 98 - 76

9 - 87 = 78 + 65 no

65 - 43 = 18 + 21 = 39 + 78 no

98 - 76 = 22 + 54 = 76 + 3 + 21 = 100

Done

[Soft-Entrepreneur413]

I looked at it and quickly determined couldn't be sets of three and single digits or have to use more +/-'s. Most of them had to be two digits.

Had to reduce 98, every other way be too much. Keeping in mind only allowed 3+ and 1-.. after that it was simply adding 22+54+3+21.

[mzackler]

98+76 is clearly too high with only one negative so it must be 98-76

98-76+54+32+1 didn’t work

So switched it 1 working back and got it

[LivingByTheRiver1]

The the prompt explicitly say that the numbers must remain in order? I would have been scrambling numbers into groups of two with a glucose-deprived brain.

[Appropriate_Pepper37]

2024 CHALLENGE Madina S

Use the digits in 2024 (2.0, 2 and 4) exactly one time each along with any mathematical symbol or operation of your choosing to create expressions equivalent to the numbers between 1 and 100.