Here's how I'd do it:

For the sixpack, I'd just substitute the cans for the quarters. So six cans, six quarters. Then I instantly notice that there are one and a half quartet of quarters.


> $1.50 $1.50
> $1.50 $1.50

Here, I'd do it much the same way as you, without the heavy visuals. :-) I'd just think of 1.50 + 1.50 as 3, and in my head I'd then have just two threes making a six. All this comes mostly from experience, though.


> What I really want to know is, how many of you are sitting there shocked and appalled, and how many of you would have done something similar?

I substitute the real stuff for numbers. In other words, I visualize the numbers, not the things.


> As an aside, another thing I tend to do is visualize multiplication problems in my head. When I do a simple problem like 24 x 7 in my head, what I actually see is:
>
> 24
> x7
> ---

I just see the numbers side by side, then I multiply seven with four and get 28, keep that in mind while I multiply twenty with four and get 140. But then I do something a bit funny: I visualize putting the two of the 28 on top of the 4 in 140, because I know that the 1 won't change and that the 0 will have no effect. (This is also something I know from experience; 40+28 won't be more than 100.) So then I know it's 168 total.

But for an assignment like this, it gets a bit trickier:

40284/3

Now, how does one go about solving something like that, with virtually no obvious "hooks" you can grab hold of?

Me, I break it down from the top, which, when I think about it, is the way it's done on paper, too, but in a slightly different way.

The largest round number being a multiple of three in 40284 is 30000. So I start off with 10000 as result, and subtract 30000 from the original number, leaving 10284. (This is done by visualizing the number 40284, fixing the last four numbers, subtracting 3 from the first 4, leaving 1, followed by the "fixed" 0284.)

10000 can be broken up into three 3000s, totalling to 9000. So I add the 3000 to the result, getting 13000, and using the above method, I fix 284 and subtract 9 from the 10, leaving 1 followed by 284 -- 1284.

Now there's a "hook" (meaning something I've learned from experience, something my mind instantly recognizes): 1200/3 is 400, so I add 400 to the result, getting 13400, leaving 84.

60 is the highest round number I can think of being divisible by three. Now I start having problems keeping all the numbers in my head, but I manage after a few moments. Result is now 13460, and I'm left with 24, which divided by three is 8, so the final result is 13468.

How would other people handle the same thing?

Trav"oooh, dizzy"holt.