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Copyright Hardy K. S. Leung

Argonaut Math Olympiad

5.9 Systematic Counting — Part I

In the classroom:

Lessons with guided exercises ( minutes)

Break ( minutes)

Lessons with guided exercises ( minutes)

Wrap up ( minutes)

Homework:

All Non-starred questions

Homework submission at

40

0

30

5

www.argomath.com/homework−−−−−−−−−−−−−−−−−−−−−−−−−−

P.2Systematic Counting — Part I

Counting problems are tricky. Counting as an the activity may not be hard, buthow do you know you didn't make a mistake, a mental error? You (usually)can't. Your best bet is to use methods that eliminate the possibility of error,and the main technique to do so is to be systematic. Previously, we learned about Fence-Post (linear) counting and circularcounting. In fact, I consider them main techniques of systematic counting:

Fence-Post counting — by always referring to the Fence-Postterminologies, you're unlikely to confuse between posts and panels. How many multiples of between and ? The first post is at , and the last is at . The length of the fence is

, and so there are panels. The number ofposts is . There are multiples of .

Circular Post counting — using transformation, we systematically convertproblems from difficult ones to easy ones without changing the answer. Today is Tuesday. What day of the week is days from today? The answer is the same if we increase or decrease by any multiple of

, for example:

Now ask yourself: what day of the week is days fromTuesday? The answer is Thursday.

5 101 199

105 195195 − 105 = 90 90 ÷ 5 = 18

19 19 5

100

1007

100 →

decrease by 70

30 →

decrease by 28

2

100 − 98 = 2


Today we'll show other ways to count systematically, and we'll name them,similar to what we did with Geometry:

Gridlines

Negative-Space

Detach-and-Reattach

Snaky-Snake

Naming techniques helps abstract the techniques into well-defined but tinytools — or spells — so you can focus on critical thinking.

Technique — Count by Categories

Instead of arbitrarily counting, we'll first find out all categories of things thatneed to be counted. We only start counting if we are sure that the categoriescover all bases!

#1


Guided Exercise

Square is drawn at the bottom. Points , and are themidpoints of the sides of the square. What is the total number of squaresof all sizes which can be traced using only the line segments shown?

What are the categories here?

big squares

small squares

big squares

small squares

Hopefully, you are convinced that these categories are exhaustive. Let's countthem:

big squares:

small squares:

big squares:

small squares:

The answer is .

ACEG B, D, F H

45∘

45∘

1

4

45∘ 1

45∘ 4

10


Guided Exercise

As shown, the "checkerboard" contains one shaded square. In thisdiagram, how many squares of any size do not include the shadedsquares?

This type of problem is error-prone. You can come up with an answer, buthow do you know it is correct? Being systematic is the answer. We'll consider squares of different sizes, andwe know only , , and squares fit.

: of them.

: we'll try to place a square at each position and see which oneworks. There are squares (each number correspondsto the number of that starts in that row).

: only .

The answer is .

Guided Exercise

Suppose Sandy writes every whole number from to withoutskipping any numbers. How many times will Sandy write the digit " "?

5 × 5

1 × 1 2 × 2 3 × 3

1 × 1 24

2 × 2 2 × 24 + 2 + 2 + 4 = 122 × 2

3 × 3 3

24 + 12 + 3 = 39

1 1002


The categories here are the tens digit and the ones digit. It is good to be ableto conjure up the following table (mentally) on demand:

Be careful what numbers are included or excluded, whether the rows are to, or to !

We'll count the number of in the tens digit separately from the number of

in the unit digit. This is actually one reason why I prefer to , so thateach row has the same tens digit. By inspection, there are s in the tens digit, and s in the ones digit. Theanswer is . Actually we "cheated" a little bit by using a table, chosen because it iseasy to count. It's okay, because neither nor have any twos, soincluding , excluding , and allowing leading zeroes don't matter.However, we should be mindful of the "hacks" that we made. If the questionis asking how many times will Sandy write " ", the answer ( ) would bevery different!

00 01 02 03 04 05 06 07 08 09

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

09 1 10

2s

2s 0 9

10 2 10 220

00 − 9900 100

00 100

0 11


Technique — Count by Casework

This technique is similar to Counting-by-Categories, except that we don'thave distinct categories, but distinct "cases".

Guided Exercise

Fifteen darts have landed on the dartboard shown. Each dart scores , ,or points. In how many different ways can the fifteen darts score a totalof points?

There are too many variations, and it is hard to correctly list all possibilitieswithout making any mistake. Instead, can we go through all variations but ina systematic way? Well, we can exhaustively consider all cases of points:

Case : s

Case : s

Case : s

It is easy to see that this is exhaustive. But how does it help? Well, say youare considering Case , with -point hits. The -point hits already accountfor points, so there are remaining hits and to make aremaining score of points. The solution, if exists, must beunique!

#2

3 5775

7

#0 0 7

#1 1 7

#2 2 7

…

#4 4 7 74 × 7 = 28 15 − 4 = 11

75 − 28 = 47


s score from s s/ s score from s/ s Solution?

We can prepare this table in a rather mechanical fashion. We don't have to gotoo far because after a while there are too many s! Now we solve it:

s score from s s/ s score from s/ s ( s, s)

—

In fact, by being systematic, even the solutions form a pattern! The answer is .

#7 7 # 3 5 3 5

0 0 15 75

1 7 14 68

2 14 13 61

3 21 12 54

4 28 11 47

5 35 10 40

6 42 9 33

7 49 8 26

8 56 7 19

7

#7 7 # 3 5 3 5 #3 #5

0 0 15 75 (0, 15)

1 7 14 68 (1, 13)

2 14 13 61 (2, 11)

3 21 12 54 (3, 9)

4 28 11 47 (4, 7)

5 35 10 40 (5, 5)

6 42 9 33 (6, 3)

7 49 8 26 (7, 1)

8 56 7 19

8


Technique — Count by Faces ( )

Each cube has faces, say facing North, East, South, West, Up, Down.Sometimes it is helpful to consider the faces individually. also take advantageof "sibling" faces with identical measurement.

Guided Exercise

Eight cubes are glued together to form the figure shown. The length ofan edge of each cube is centimeters. The entire figure is covered inpaint. How many square centimeters are covered in paint?

North and South: squares each. East and West: squares each. Up and Down: squares each. Hole: squares. Total: squarecentimeters. (Note: in this case it is more convenient to count the hole separately, but in someother cases, you can also attribute the hole to each of the six directions)

#3A 3D

6

3

8

3

3

4

(8 + 8 + 3 + 3 + 3 + 3 + 4) × (3 × 3) = 32 × 9 = 288


Guided Exercise

Twenty unit cubes are glued together to form this figure, with "holes"which you can see through. The total figure measures . If thefigure is fully dipped in a bucket of paint, how many square units ofsurface area would be painted?

You can observe that all faces are identical to each other, so we'll only need tocalculate the area of a single face, say the UP face. Be careful with the squares that are facing UP but hidden inside the interiorof the cube. are in the interior, and are on the surface, for a total of squares. Since the cube, like a die, has faces, the answer is .

3 × 3 × 3

4 8 126 6 × 12 = 72


Technique — Count by Layers ( )

Guided Exercise

The set of stairs shown at the right is constructed by placing layers ofcubes on top of each other. What is the total number of cubes containedin the staircase?

This is easy to count by layers: , , , and . The answer:

Technique — Count by Columns ( )

Guided Exercise

The tower shown is made of congruent cubes stacked on top of eachother. Some of the cubes are not visible. How many cubes in all are usedto form the tower?

#3B 3D

1 × 3 3 × 3 5 × 3 7 × 3

3(1 + 3 + 5 + 7) = 3 × 16 = 48

#3C 3D


We can solve this problem counting by layers, or by columns. Either wayworks but in this case it is easier to count the columns:

Technique — Count by Tree

Guided Exercise

Three circular streets intersect at points , and as shown.How many different paths can be walked along the streets from to ,if no intersection is entered more than once when walking each path?

This must be an all-time most diffcult problem in Math Olympiad — only 1%of students answered this correctly. To get this right, you must be verydiscipline and systematic. We will use a tree which is helpful in enumeratingoptions.

10

4 + (3 + 3) + (2 + 2 + 2) + (1 + 1 + 1 + 1) = 4 + 6 + 6 + 4 = 20

#4

A, B, C, D, E F

A F


Starting from , we repeatedly branch out to all possibilities.

We'll continue to expand each node while keeping track of the constraint weneed to satisfy (no repeated visit).

By being systematic, the solution is more robust because you may findpatterns in the system:

Finally,

A


To be careful, we'll circle the ones that are real endpoints:

There are paths.28


Guided Exercise

Numbers such as or have their digits in decreasing order becauseeach digit is less than the digit to its left. The digits in are not indecreasing order. How many whole numbers between and havetheir digits in decreasing order?

Let's use a tree:

Careful, some of the intermediate paths lead to nowhere, and we must becareful not to consider them (this is a common mistake). I prefer to circle thecorrect ones (the answer is ):

543 531322100 599

20


Technique — Count by Scratch Table

Guided Exercise

Assume that a post office issues only ¢ and ¢ stamps and all postage isin whole numbers of cents. What is the greatest amount of postage incents which cannot be made using only ¢ and ¢ stamps?

You can use trial-and-error to find out what postage can be made. But whatabout those that cannot be made? There is no obvious method to count all possibilities using some kind offormula. Yet, we still do this systematically. Sometimes a tree works better;sometimes a table does. In this case, we use a table of numbers, and we circle, or "scratch-off",numbers that satisfy our criteria. I call this a Scratch Table. It is the most usefulwhen you can only find the result by looking at the big picture after the fact,e.g. if you need to count the unique number of ways to do something, or tofind the largest unscratched number. An ancient technique used to findprime numbers, called the Sieve of Eratosthenes, is based on Scratch Table.

We start by circling (or boxing) and because we know they are "reachable"(can be made with ¢ and ¢ stamps):

#5

3 8

3 8

3 83 8


Next, we'll go down the numbers one by one, and ask, is this numberreachable by adding a ¢ stamp or a ¢ stamp to a previously-reachablepostage? If the answer is no, we'll leave it alone. If the answer is yes, we'llcircle (or box) the number to mark it as "reachable".

¢: —

¢: —

¢: Reachable (original)

¢: —

¢: —

¢: Reachable (from )

¢: —

¢: Reachable (original)


¢: —



¢: —










At this point, the table looks like this:

1 2 4 5 7 10

13

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

3 8

1

2

3

4

5

6 3

7

8

9 6

10

11 8

12 9

13

14 11

15 12

16 8

17 14

18 15

19 16

20 17

21 18

22 19

3 6 8 9

11 12 14 15 16 17 18 19 20

21 22 …


Now observe that there is a long sequence of reachable numbers startingfrom . It is not hard to reason that anything afterwards is reachable:

Therefore, we have seen the last of the uncircled, which is ¢!

14

14, 15, 16⟶ 17, 18, 19

17, 18, 19⟶ 20, 21, 22

20, 21, 22⟶ 23, 24, 25

23, 24, 25⟶ …

13


Guided Exercise

Consider all pairs of counting numbers whose sum is less than . Thetwo members of a pair could be either the same as each other ordifferent. How many different products are possible if the two numbersare multiplied?

Do you realize that the multiplications that we want are just the upper-leftcorner of the multiplication table? We can further cut it roughly byhalf because of duplicates (e.g. ):

dup —

3 dup dup — —

4 dup dup dup — — —

5 dup dup dup dup — — — —

6 dup dup dup dup — — — — —

7 dup

We need to eliminate duplicates (e.g. ), either using an actualScratch Table, or just keep track of unique numbers in-place (more error-prone).

11 13 17 19

22 23

The answer is .

11

10 × 103 × 4 = 4 × 3

× 1 2 3 4 5 6 7 8 9

1 1 2 3 4 5 6 7 8 9

2 4 6 8 10 12 14 16

9 12 15 18 21

16 20 24

25

…

2 × 6 = 3 × 4

1 2 3 4 5 6 7 8 9 10

12 14 15 16 18 20

21 24 25 …

19

P.20Systematic Counting — Part I (HW4)

Problem

The counting numbers are written out as one long string:

What is the digit in the string?

Problem

The only way that can be written as the sum of different countingnumbers is . In how many different ways can be writtenas the sum of different counting numbers? Assume that order does notmatter.

Problem

Cheryl traces her name, CHERYL, by following the lines shown. She canchange direction only at a letter. How many different paths can trace hername?

01

123456789101112 …

100th

02

10 41 + 2 + 3 + 4 15

4

03


Problem

A large rectangle is cut into smaller rectangles. How many rectanglesof all sizes are in this diagram?

Problem

Each of the boxes in the figure at the bottom is a square. How manydifferent squares can be traced using the lines in the figure?

04

6

05


Problem

An acute angle is an angle whose measure is between and . Usingthe rays in the diagram, how many different acute angles can be formed?

Problem

The complete outside including the bottom of a wooden inch cube ispainted red. The painted cube is then cut into inch cubes. How manyof the inch cubes do not have red paint on any face?

06

0∘ 90∘

07

41

1


Problem

Eight one-inch cubes are put together to form the T-figure shown at thebottom. The complete outside of the T-figure is painted red and thenseparated into one-inch cubes. How many of the cubes have exactly fourred faces?

Problem

A wooden block is inches long, inches wide, and inch high. Theblock is painted red on all six sides and then cut into sixteen inchcubes. How many of the cubes each have a total number of red faces thatis an even number?

08

09

4 4 11


Problem

The figure shown consists of layers of cubes with no gaps. Suppose thecomplete exterior of the figure (including the bottom) is painted red andthen separated into individual cubes. How many of these cubes haveexactly red faces?

Problem

The tower at the bottom is made up of five horizontal layers of cubeswith no gaps. How many individual cubes are in the tower?

10

3

3

11


Problem

How many two-digit numbers are there in which the tens digit is greaterthan the ones digit?

Problem

At the right is a by by cube. Not all of the cubes are visible.Suppose the entire outside of the cube is painted red including thebottom. How many different by by cubes with exactly three redfaces can be formed in the shown cube?

12

13

3 3 3

2 2 2


Problem

Each of the small boxes in the diagram at the bottom is a square andcongruent to each of the others. How many different squares can betraced using the lines of the diagram as sides?

Problem

There are exactly six different three-digit numbers that can be formedusing each of the digits , , and exactly once in each number. Find theaverage of these six three-digit numbers.

14

15

4 5 6


Problem

The tower shown at the bottom is made of unit cubes stacked on top ofeach other. Some of the unit cubes are not visible. How many unit cubesare not visible?

Problem

Three darts are thrown at the dartboard shown. A miss scores points.The three scores are added together. Find the least whole number totalscore that is impossible to obtain.

16

17⋆

0


Problem

is a triple of counting numbers which has a sum of . Consider and to be the same triple as . How many

different triples of counting numbers have a sum of ? Including as one of your triples.

Problem

Find the sum of the digits of the first odd counting numbers.

18⋆

(1, 1, 8) 10(1, 8, 1) (8, 1, 1) (1, 1, 8)

10(1, 1, 8)

19⋆

25

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