Expert: Scott A Wilson - 5/1/2009

Question
My 5th grade daughter comes home with a weekly brain teaser for her advanced math class.  This week's has even my very mathmaticly talented husband stumped.  Can you help us out?
The questions are:

These are:  28,96,48,60,12   These are Not: 300,90,26,47, 50  Then which of these are?  16,100,25,52,38
Because.....

These are: 615,57,291,804,93,  These are Not: 64,913,562, 215,1923  Then which of these are? 325,417,273,86,66  Because.....

These are: 1,3,15,5,9, Are Not:  8,7,6,10,11  Then which of these are? 2,12,45,22,4  Because......

Answer
These are:  28,96,48,60,12   These are Not: 300,90,26,47, 50  Then which of these are?  16,100,25,52,38;  Because.....

That may have taken me awhile to look about, but the answer was revealed to me after a little thought now and then.

1.My first thought was that
28 = 2^5 - 2^2,
96 = 2^7 - 2^5,
48 = 2^6 - 2^4,
60 = 2^6 - 2^2,
12 = 2^4 - 2^2.

From what I can see, 300, 90, 26, 47, and 50 are not the difference of two powers of two.

Out of the numbers in the list, 16 = 2^5 - 2^4 = 32 - 16,
so thats in the list.

But then I thought of an even simpler way, rather than trying to show the ones not it in list were not a difference of two powers of two.

The numbers in the list are all numbers between 1 and 25 that have been multiplied by 4.  Of the numbers not in the list, 300 is the only one divisible by 4.  However, divide 300 by 4 and you get 75, which is above 25.

Of the ones in which you have the choice of adding, add 16, 52, and 100, for they are, respectively, 4*4, 4*13, and 4*25.

If you want a good excuse for it being 1-25, at 26, when the numbers are squared, the last two digits start the sequence backwards.
When 1-25 are all squared, you get 1, 4, 9, 16, 25, 49, 64, 81, 100, 121, 144, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625.
Note that the last digits give you the series 01, 04, 09, 16, 25, 36, 49, 64, 81, 00, 21, 44, 68, 96, 25, 56, 89, 24, 61, 00, 41, 84, 29, 76, 25.

At 25, the series of the last two digits repeats backwards.
That is 26² = 676, and 76 ishow 24² ends (576).  27² = 729, and 29 is how 23² end (529).  28² = 784, and 84 is how 22² ends (484).

I'm not sure this is right, but it look like a good way to do it.


These are: 615,57,291,804,93,  These are Not: 64,913,562, 215,1923  Then which of these are? 325,417,273,86,66  Because.....

For the second list (615,57,291,804,93,), they are all lesser numbers times 3.  That is, take 205, 19, 97, 268, and 31, each times 3.  The result is the list given, in order.

Of the number 64, 913, 562, 215, and 1923, none are divisible by 3.

The numbers to add are 417, 273, and 66.  They are, respectively, 3 times 139, 91, and 22.


These are: 1,3,15,5,9, Are Not:  8,7,6,10,11  Then which of these are? 2,12,45,22,4  Because......

The numbers 1, 3, 15, 5, and 9 are all multiples of the first 3 odd numbers, 1, 3, and 5.  1=1, 3=3, 15=3*5, 5=5, 9=3*3.

Of the numbers   8,7,6,10,11 , none are multiples that involve all 1's, 3's, and 5's.  They can all be said to have a 1, but there are other number that aren't 1, 3, or 5.  Note that 6 is divisible by 3, but it is also divisible by 2, which is not included.  10 is divisible by 5, but it is also divisible by 2, and that is not included.




I'm not sure this meets the answers they give,
but it looks good to me.