# Department of Science, Math, & Technology

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Survival puzzle
Posted: Posted October 7th, 2017
Edited October 7th, 2017 by EN
 Getting lost while hiking in the wilderness is a dangerous situation to find yourself in. And making your way back to civilization is a difficult task that quickly uses up resources. What you decide to take with you while making the journey back to civilization can determine life or death. The picture shows all of the items that are available to you that will aid you in your hike out of the wilderness. Containers of Food and Water will give you energy, Shelter will protect you from the elements, and Defense will protect you from wild animals. Each item has a weight indicated by the red number and each item has survival points indicated by the green number. You must take only one item from each of the four categories (Food, Water, Shelter, Defense). Unfortunately, the backpack you have has a maximum capacity of 25 kg. Your chance for survival is calculated by adding all of the survival points together from the items you choose to take with you. What is the maximum chance for survival you can achieve? How do you achieve it? I'll post my answers and explanation early next week.
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There are 5 Replies

Well, I divided all the survivals by the weights and picked the highest number in all categories -- that ended up being the middle row, with a weight of 23 and a survival score of 70. But I can do better.

Posted October 8th, 2017 by Xhin
Xhin
The planets are aligned

At this point my only move is to pick the biggest stick, which makes my weight 24 and my SS 75.

this is probably the most optimal configuration because the value of all categories per weight is the highest, aside from the wood, where I add more survival points for a minimum increase of weight.

Posted October 8th, 2017 by Xhin
Xhin
The planets are aligned

Hey that sounds sort of like a Lagrange multiplier problem in 4 (discrete) variables $$x, y, z, w$$ where $$x$$ is one of the kg weighs of food, $$y$$ one of the kilograms of water, etc.

The constraint in the problem would be

$$x + y + z + w \le 25$$

subject to which you want to maximize the survival function

$$S = f(x) + g( y) + h(z) + j(w)$$

where $$f(x), g( y), \dots$$ are the survival points associate to $$x, y, \dots$$. For example, $$f(5)=10, f(8)=20, g(3)=10, h(8)=15, \dots$$ etc.

Edited October 8th, 2017 by The Fly

The maximum survival number is 90 when you pick up the highest kg in each category, but which exceeds the 25 kg limit. So the answer will be less than 90. But since all the survival points are multiples of 5 (so you can actually divide them all by 5 to make it simpler), the next possible survival scores are 85, 80, and 75 which Xhin showed is achievable. So we are only left with showing whether 80 and 85 is achievable or otherwise with the 25 kg constraint imposed. I suspect 80 and 85 aren't consistent with the 25 kg limit, so 75 looks to be the one.

Edited October 8th, 2017 by The Fly

That's the answer I got, Xhin. Deduction is one way to solve it, as The Fly started to show. Looks like Lagrange multipliers is another method, although I'm not very familiar with it.

I used Linear (Integer) Programming to solve this problem. I found an LP solver package for Python, translated the problem into a form the program can handle, and got the answer in a split second.

I'll post another problem (at least one) along the same lines as this one that's a bit harder. But it might be relevant to all of the games you run around here.

Posted October 10th, 2017 by EN