Problem of the Month
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February's Problem

Find the sum of the DIGITS of the first 100 ODD NUMBERS.

A solution to this problem

will appear along with next month’s problem.

 

 

 

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
 

 

For many additional problems we highly recommend the following books:

Math Olympiad Contest Problems Volume 2 edited by Richard Kalman

and

Math Olympiad Contest Problems for Elementary and Middle Schools by Dr. G. Lenchner are sources of many such problems.

Creative Problem Solving in School Mathematics 2nd Edition by Dr. George Lenchner can help you to teach solving these types of problems