|
Solution
to January's Problem
|
|
The
first ten numbers in a sequence are
1,
2, 2, 3, 3, 3, 4, 4, 4, 4, ....
What
is the 500th number in the sequence?
|
|
ANSWER:
32
Solution:
Consider
this table representing the one "1", the two "2"'s,
the three "3"'s, etc and compare the last position
for each number to the sum of the numbers 1+2+3+...+the number
|
number
|
repeats
of number
|
positions
for number
|
sum
1+2+...+number
|
|
1
|
1
|
1
|
1
|
|
2
|
2
|
2,3
|
3
|
|
3
|
3
|
4,5,6
|
6
|
|
4
|
4
|
7,8,9,10
|
10
|
|
5
|
5
|
11,12,13,14,15
|
15
|
|
.
. .
|
.
. .
|
.
. .
|
.
. .
|
|
N
|
N
|
.
. .
|
N(N+1)/2
*
|
|
.
. .
|
.
. .
|
.
. .
|
.
. .
|
|
30
|
30
|
436,...,465
|
465
|
|
31
|
31
|
466,...,496
|
496
|
|
32
|
32
|
497,...,528
|
528
|
That
shows the number 31 appears in positions 466 through 496 while
32 appears in positions 497 through 528. The 500th number
is therefore 32.
|
|
*
Recall: The sum 1 + 2 + 3 + 4 + ... + N = N(N+1)/2 which
can be demonstrated as follows
Row A is
the sum written out. Row B, the same sum rewritten in reverse.
Add rows A and B to get row C. Row D is row C written as a product.
Divide both sides of the equation by 2 to arrive at row E.
|
|
Solution
to December's Problem
|
| A
10x10x10cube is painted, then cut into 1000 1x1x1 cubes.
How many of these cubes are painted on exactly 2 sides?
|
|
|
|
ANSWER:
96
Solution:
The
cubes painted on 2 sides (faces) are the 8 interior cubes along
each of the 12 edges. 8 x 12 = 96
Note:
Consider
all 1000 cubes
|
number
of painted faces
|
number
of cubes
|
position
of these cubes
|
|
0
|
8x8x8=512
|
The
8x8x8 cube inside the 10x10x10 cube. Strip away the outer
one layer from each face.
|
|
1
|
(8x8)x6=384
|
The
8x8 square that is the interior of each of the 6 faces
|
|
2
|
8x12=96
|
Each
edge except for the 8 corners
|
|
3
|
8
|
The
8 corners
|
|
4
|
0
|
Each
cube is adjacent to at least 3 oather cubes
|
|
5
|
0
|
|
|
6
|
0
|
|
|
TOTAL
|
1000
|
|
|
|
