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1.

Consider a 5x3 rectangle- it is 5 unit squares wide and 3 unit squares tall, thus made up of 15 little squares. If you shade the outside squares, the border squares, 12 squares are shaded, with 3 in the middle left unshaded.

Can you make rectangle with its border squares shaded so that the number of shaded squares equals the number of squares in the center? If so, how many rectangles (including squares) like that can you make? What are the dimensions of those rectangles?

If you think for a bit, you will realize that the formula for the amount of squares on the edge of a rectangle is:

border squares shaded = 2x+2y-4

(the -4 to eliminate the 4 overlapped corner squares)

where x and y are the dimensions of the rectangle

2.

A guy was having a paper written in English translated into French. He got the assistance of a French translator named Jacques. At the bottom of the paper was the following: (written in French of course)

"Much thanks to my friend Jacques for translating the above paper into French."

and then:

"More thanks to my friend Jacques for translating the above sentence into French."

and then:

"More thanks to my friend Jacques for translating the above sentence into French."

At first glance, this would have to continue forever, for proper thanks to be due. But it doesn't need too... it can (and does) end right there. Why?

no hint, this is easy. Use your brain

:P
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Yep, and to brhan: there are number bases based off of irrational numbers, but they're kind of complicated lol. My favorite is phinary, just cuz the Golden Ratio (and fibonacci numbers) are so amazing

anyway, how I would mathematically solve #3 if I didnt already know the answer:

10101 in base x is, in other words:

x0 + x2 + x4 in base 10

or

1 + x2 + x4 in base 10

changing 273 from base x+6 into base 10 is also easy:

2*(x+6)log(100) + 7*(x+6)log(10) + 3*(x+6)log(1)

or:

2*(x+6)2 + 7*(x+6)1 + 3

now that they are both in base 10, they can equal each other:

1 + x2 + x4 = 2*(x+6)2 + 7*(x+6)1 + 3

subtract 1 from both sides

x2 + x4 = 2*(x+6)2 + 7*(x+6)1 + 2

(x+6)2 is (x+6)(x+6), which is x2+12x+36, all multiplied by 2 is 2x2+24x+72

7*(x+6)1 is 7x+42

x2 + x4 = 2x2+24x+72 + 7x + 42 + 2

now we have a ton of like terms on the right side to merge:

x2 + x4 = 2x2 + 31x + 116

subtract x2 from both sides

x4 = x2 + 31x + 116

subtract x4

0 = -x4 + x2 + 31x + 116

*-1

0 = x4 - x2 - 31x - 116

factor

0 = (x-4)(x3+4x2+12x+17)

From there, what can easily turn that into 0. The clear answer is 4, cuz 4-4=0, which would negate the rest of it. If you try that, 10101 in base 4 is 273 in base 10 (4+6)

I believe this was brhan's method. This is the mathematical/algebraic way to do it.

The logic way to do it, which is probably what I would have really done, is:

* x+6 cant be 7 or smaller, cuz a 7 appears in the number so it would carry over as a 10 if x+6 was 7 or lower. Therefore x+6 is 8 or more

* which means x is 2 or more

* x+6 could be more than 10, but no symbols were used in 273 so we're not sure if x+6>10 yet, though it's unlikely, since no symbols were used in 273, so it's not showing the part of the base, and what symbols would you use, letters..? To see if that's an option, figure out if x was higher than 4 and see if that's a possible solution. However if x was higher than 4, for example 5, than 1+5^2+5^4 = 651 in base 10, or, in base 11, 542, already much too big than 273

* so our range is x=2 or 3 or 4

* if x was 2, simple binary, then x^4+x^2+x^0 is 16+4+1 = 21 in base 10, which is (16=20)+5 = 25 in base 8 (8=x+6 if was x was 2). Clearly 25 is much less than 273.

* I could try x=3, but I can already see that 1+9 is 10 in base 10, or 11 in base 9, so 3^4 would have to be 262 in base 9, but I know that 3^4 is 9^2 and 9^2 is 81 in base 10- which is very cleanly 100 in base 9. 100+11=111 in base 9, not 273. Not even close. So I dont even need to check base 3 fully

* the only thing left is x=4

* 1+16+256 is 273. Yay! The number! But wait- dont you have to convert it to base 10, so it will change *sinking feeling* But not! Because if x is 4, than x+6 is 10. 273 is in both base 10 and base x+6.

Therefore x=4

:P

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