Mathematical problems are so much easier if you cheat
Mind Your Decisions is a YouTube channel of brain teasers, riddles, logic puzzles, viral math problems, and topics from viewers around the world
, presented by Presh Talwalkar. Many of the puzzles are readily solved by Indian and Chinese schoolchildren, just to make me feel good about myself. (rolleyes)
In How To Solve The 6s Challenge, he presents eleven incomplete equations:
The challenge is to complete and balance the equations using only common mathematical operations. There are two stipulations: firstly, the introduction of new digits is not permitted; and, secondly, the result must equal 6, other comparators are not allowed. For the first stipulation he specifically cites, as an example, that the cube root of a number is not allowed, since it introduces an additional digit, 3, like so ∛.
Some of the solutions are so obvious that even I could work them out:
So far, so good. But what about 9 9 9 = 6? Well, apparently, that's simple, because the square root of 9 is 3, and we've already solved the problem for 3:
Wait a minute, though, we're not allowed to introduce any more digits! Strictly speaking, unlike the cube root the symbol for square root, √, doesn't introduce a digit, but that's only by convention. We can exclude 2 from the symbol, since the square root is the base root and 2 is implied by default. But it is there, whether explicit or implicit.
This can be more readily understood if roots are expressed as power functions. The nth root of any number is simply that number raised to the reciprocal of n. Thus, the cube root of a is a raised to the power of one third:
Similarly, the square root of a is a raised to the power of one half:
Or, if you prefer:
Clearly then,
Consequently, any solution relying on the square root function is implicitly introducing a new digit, and thus contravenes the first stipulation. This is cheating.
The prosecution rests its case, m'lud.
I don't claim unique credit for this observation, although it did occur to me while watching the video, and before delving into the comments. Previous commentators have also pointed out this flaw.
According to at least one critic of this argument:It said digit, not number. So it [square root] is allowed since he wrote the square root. If that rule applied to everything, then multiplication wouldn’t be allowed either because that technically counts as this example: 2 x 2 x 2 = 8 (2 + 2 + 2 + 2 = 8) And that does add another number, but the rules said digits so it is allowed.
What our fearless bonehead—who seems oblivious to the fact that numbers are constructed of digits, but √ is neither—overlooks is that while multiplication has the same effect as multiple successive additions, this involves a change in mathematical operator. Extra digits are not implied by the × operator, and it can operate without invoking multiple + operations. Unless bonehead actually would manually calculate 180 × 15 by adding 180 to itself 14 times. (SMH)