This came across on Facebook, and someone said he needed a formula or something to help him solve this kind of puzzle. Matthew Eastland, the son of an old friend of mine pointed out that there are apparently two, or maybe 3 solutions that are possible. I composed an answer to a friend of a friend that had picked one of the "incorrect" answers but messed up one step. Here are my thoughts.
#1 Take it at face value that only the first equation is correct. That is, we know that 1+4=5. The others are nonsense statements, so 8+11=19.
#2 Taking the four equations as separate and equal, you look for a formula that will make each correct. In such, you look at each number and check the relationship with the others. In the second case, what do you add to 2 to get 12? What does that number have in common with the 5? You might ask the question like this:
If 1+4=5 and 2+5=12 and 3+7=21, then what does 8+11 equal? You apply the same rules to each separate equation, rather than to the string as connected problem. The answer this way is 96. The formula is x+xy=z.
You might notice that the first three equations form a progression with a twist. x goes from 1 to 3; and y goes from 4 to 7. THEN, x increases to one more than y in the previous equation in equation #4. This leads in a way to choice #3.
#3 The answer of 40 comes by reading the problem, looking for a relative position of each of the integers rather than looking for a formula in the strictest sense. By what is probably not coincidental (I have not done the math behind it), a relationship is found between all the integers shown.
Assuming the four lines as parts of the whole, and for convenience, make each number a signed integer.
+1+4= +5 +2+5 = +12 +3+6= +21 +8+11=?
In this way, each part builds on the other one. The last equation becomes 21+8+11= 40.
A closer look at equations 1, 2 and 3 show a progression of the "answers" by the odd numbers of 7, 9 and 11. This is a clue that there is an algebraic progression, pointing to #2 being the correct answer. However, if we don't treat each line uniquely, then #3 moves us to look for a linear progression instead. That is why I lined the equations up the way I did (sort of like taking away the "wordwrap."
It seems like the analysis depends on the approach to the "truth." #1 takes it at its face value, leading one to reject obvious falsehood. Let's face it, 2+5 =/= 12. Open and shut case! Everyone knows that 8+11 = 19!
But assuming that there is more to the equations than meets the eye, the integers take on different meanings in solution #2. Each is taken to represent the same principle. Some formula has to be found to make each statement true. And then that principle is applied to make the last statement true as well. This is like comparing different testimonies in a court of law.
Solution #3 makes correlations between the separate integers and signs. Each step is taken based on the symbol (assumed or present) to its left. Like #1, everything is taken at face value. All the known facts are laid out and the observer builds a story out of them. I call this "linear" thinking. When the equations in the middle seemed like nonsense, their context was used to arrive at a conclusion.
Different approaches, in the real world outside of numbers, can lead to solutions to big problems. Assuming facts to be false will leave many a mystery unsolved. Likewise, using known facts out of context will lead to wrong conclusions. It is only by fully analyzing the evidence can we come to the solution of the problems we face.
Next week, I may be chosen to sit on a jury. In a court of law, it is necessary for a jury, or a judge, to apply logic that goes deeper than the "face value" of the facts. Bias cannot be allowed whereby evidence is cast aside due to it's seeming nonsense. Some things we "know" might not be true. Just like in solution #2, all is not as it seems. Likewise, as in solution #3, facts out of context can lead to jumping to conclusions.
It is only through analytical thinking that progress towards truth can come. Let us not "jump to conclusions" or make "snap judgments.
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