If you have studied some of the old SAT questions at some point you may have gone through questions like –
4 consecutive numbers sum to 166. What is the product of the numbers? or…
3 consecutive even numbers sum to X. What is their product?
The way the tutors and the online tutorials show seems to always be to algebraically solve this – for example 4 consecutive numbers would solve x+x+1+x+2+x+3 = 166, collect terms and solve…
However there is another way that works for this and works for other similar problems.
First of all consider all the numbers to be the same. Now 4x = 166 and the number would be 41.5, this is directly between the two middle numbers and the four numbers are 40,41,42,43. Indeed this is the answer! If you were looking for sum of 3, 5, or 7 consecutive numbers, the number would be the middle one, not between the middle ones.
I also found this to work in general even for some more complicated problems. For example what consecutive numbers a,b,c,d have a value of a2+b2+c2+d2 = 6894? Divide by 4 then square root, as if all numbers were the same, then you will find 41.515…, and in fact the four numbers are the same as the ones in the previous answer. 40,41,42,43.
I tried a variety of different formulas in which calling them the same number, works for consecutive numbers. In fact, for some methods like sum of consecutive numbers, you can represent the “middle” number as x and prove this:
(x-1.5)+(x-.5)+(x+.5)+(x+1.5) = x+x+x+x.
The sum of these when considering an odd number of terms, is the same problem, and can be considered to all be the same, divide by how many numbers to get the average:
(x-2)+(x-1)+x+(x+1)+(x+2) = x+x+x+x+x
and for 3 consecutive even numbers’ sum, this would be
(x-2)+x+(x+2) = x+x+x
which also cancels out to three times the middle number. While looking at this pattern it may be tempting to consider this proven for these cases, but not so fast…
What about the squared case for the numbers:
-12+02+12+22 = 6
A negative squares to a positive and this changes the expected middle to be:
sqrt(6/4) = 1.2
yet that value is wrong:
02+12+22+32 = 14
If you look at x2 graph you will see it is not strictly increasing, so it makes sense that this happens for cases like this.
Product
Does this trick work for a product? Let’s try:
10*11*12*13 = 17160
Now let’s consider this all the same, x*x*x*x=17160 has a solution 4th root of 17160 (or 1/4 power if you are entering this in a spreadsheet, “=17160^.25”.
The answer, 11.44, is between 11 and 12! It works! It works for many other numbers as well. However, for:
0*1*2*3 = 0
and this trick doesn’t work all the time 🙁
Another try – square roots
What about square root? That is strictly increasing and if you try the same thing it appears to work for just about any numbers:
sqrt(1)+sqrt(2)+sqrt(3) = 4.15
Considering them to be same, sqrt(x)+sqrt(x)+sqrt(x)=4.15, divide by three:
sqrt(x) = 1.38
square it and you get x=1.9, closest to the integer value of 2, the correct middle number. Try that again for other numbers, larger numbers and the rule seems to work even better:
sqrt(21)+sqrt(22)+sqrt(23) = 14.06,
divide by 3 and square to get 21.99.
Try again for four and it seems we have a good pattern:
sqrt(0)+sqrt(1)+sqrt(2)+sqrt(3) = 4.146
divide by 4 and square to get 1.07, a number between 1 and 2. Try this for any larger numbers and the pattern appears to get even better:
sqrt(47)+sqrt(48)+ssqrt(49)+sqrt(50) = 27.85
divide by 4 and square to get 48.49, more closely exactly between 48 and 49.
So have we found another general pattern we can use? This seems to work for 5 terms as well, resulting in a number closest to the middle whole number (but not exactly equal). Not so fast, though…
sqrt(0)+sqrt(1)+sqrt(2)+sqrt(3)+sqrt(4)+sqrt(5)+sqrt(6)=10.83
Divide by 7 and square, you get 2.39 which is not nearest the middle whole number (three here). (Oddly enough the larger numbers do all seem to work here, continuing to get closer and closer? This is interesting and might be an interesting further investigation?)
Conclusion
This goes to show that when you do find an interesting pattern in numbers, if it is not a formal proof you are likely to find some counterexample disproving the trick… unless you have a proof! as with the summation of consecutive numbers – the algebra-proved first math trick above. A math observation can be just that, a math observation, without a proof – which is the field that many of the prime conjectures now live in until proved or disproved. The result of machine learning and neural nets is also often like that in that it works for a majority (or at least a majority of tested) cases, yet can be mistaken in a variety of data points. Bugs in Google Translate, for example. So if you have a “math trick” that does not have a formal proof (or similarly, a neural net that seems to do what you think it will)… be prepared for some odd erroneous cases.
Lastly, if you are doing one of those tricky SAT problems you should look back at the question – is the answer the product or sum or the largest of the numbers, that it is asking for? And check the answer in the original question.