Worst. Hint. Ever.

On December 6th 2014, freelance writer Elna Cain entitled a blog post,

The Worst Thing You Can Do on a Blog Post (Hint: You are Probably Doing it)

This has to be one of the worst (or at least most irritating) hints in the history of bad hints. The things I am currently doing while composing this blog post are sitting, typing, drinking tea, and cursing the boys who are ringing my doorbell trying to get back into the house three hours before the end of the leave weekend. None of these things strike me as the worst thing I can do on (in?) a blog post.

Scrolling down the article, I discover that the worst thing I can do is "not crafting an attention getting, eye-catching, interest grabbing headline".

I pause to ponder the irony.

I pause again to wonder why attention-getting and interest-grabbing have not been hyphenated.

I move on.

I mention all this because I came across an even worse hint this weekend. One of my colleagues (who shall remain nameless) was struggling with the maths homework that his six year-old daughter had been sent home with. It was sent to me in an email entitled, "Is this even possible?". Now that's what I call an interest-grabbing headline!

If you wish to try this problem yourself, scroll no further

The problem seems a simple one, although there are 3024 ways of inserting the digits into the grid and a young child might get bored before having checked all of them.

It's a good job there's a hint at the bottom. Let's see...

"You may find it helpful to arrange the four two-digit numbers in columns to do the addition"

No need. I have a calculator. Next...

"and think about what digits you need to get a 0 in the units"

Clever. This must really cut down the number of possible permutations.

In order to make things a little organised, we'll call the entered digits a, b, c, and d as follows:

Now the only way to get a last digit of 0 in the sum's total is to get 10, 20, or 30 when I add b, c, d, and d again. For example, we could have b as 3, c as 7, and d as 5: 3+7+5+5=20. Then we have six possibilities for a and we need to see if any of these solves the problem and gives 100. The hint does say, "and then look at the tens" but you need to know what a is to do that anyway. You might as well tot up the lot and move on if you are unsuccessful.

So does this hint cut things down a great deal? Well, yes, but only after a brute-force and rather tedious search. Furthermore, it still leaves 276 possible permutations which is a lot for a six year-old to check. Can they even count to 276 by that age?

The hint is rubbish. A far better way of tackling the problem is from a onwards not from d backwards. The four two-digit numbers we get are ab, ac, bd, and cd. They need to add to 100. If we were writing this algebraically we would find that

or, more simply,

The coefficient of a is quite interesting. It is so big that if a is 4,5,6,7,8, or 9, the sum will exceed 100 whatever we make b, c, or d.

In fact, if a is 3 then the remaining three terms must sum to 40. This means that b and c must be 1 and 2 and so 2d would have to equal 7. Since d is an integer, this is also impossible.

Almost immediately we are down to only two possible values of a and a total of 672 possible combinations. This is an anagram of the 276 arrangements discussed earlier. Coincidence? (Yes, probably.)

So what if a is 2? This means that the remaining three terms must sum to give 60. Now the total of the b and c terms will be a multiple of 11 and, in fact, must be an even multiple of 11 (as the d term must be even). This means it must be 44 and we get the solutions:

Trying a=1 yields another mirrored pair:

And that's it. That is all there are. This deduction was made in far less time than it took to search for all the combinations that gave a final digit of zero. It is actually a reasonably interesting question if the hint is ignored.

Worst. Hint. Ever.

I can only imagine that the author of the problem had some children that needed to be kept quiet for several hours. With that in mind, I might give this problem to some of my adolescent students this week. It's a good job that they are all far too cool to be spending their weekends reading a maths blog.

from matheminutes http://ift.tt/2dBPECq