Shuli s Math Problem Solving Column

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1 Shuli s Math Problem Solving Column Volume 1, Issue 18 April 16, 009 Edited and Authored by Shuli Song Colorado Springs, Colorado shuli_song@yahooom Content 1 Math Trik: Mental Calulation: 19a19b Math Competition Skill: Systematially Listing Aording to Shapes 3 A Problem from a Real Math Competition 4 Answers to All Pratie Problems in Last Issue 5 Solutions to Creative Thinking Problems 46 to 48 6 Clues to Creative Thinking Problems 49 to 51 7 Creative Thinking Problems 5 to 54 Math Trik The Trik Mentally alulate: In eah multipliation the two numbers are lose to 00 Write the multipliations in the general form: 19a19b where a and b are digits Let 0019a and d 0019b Then 00 d 19a19b 00 The short ut is shown through the following examples Example 1 Mental Calulation: 19a 19b Calulate Step 1: Calulate 00 19a and d 00 19b In this example, and d Step : Calulate 19 a d or 19 b In this example, 19a d or 19b Step 3: Calulate (19a d) or (19b ) In this example, Step 4: Calulate d In this example, 7 14 Step 5: Attah the result in step 4 as two digits to the right of the result in step 3 In this example, attah 14 to the right of 38: 3814 Now we are done: Example Calulate Step 1: Calulate and Step : Calulate or Step 3: Calulate Step 4: Calulate 3 6, treated as two digits: 06 Step 5: Attah 06 to the right of 390: We have This works for two numbers not so lose to 00 Example 3 Calulate Step 1: Calulate and Step : Calulate or Step 3: Calulate Step 4: Calulate Step 5 Attah 77 to the right of 364: We obtain Example 4 Calulate Step 1: Calulate and Step : Calulate or

2 Shuli s Math Problem Solving Column Volume 1, Issue 18 April 16, 009 Step 3: Calulate Step 4: Calulate Step 5: Add 1 to 350 yielding 351, and attah 54 to the right of 351: We have Why Does This Work? 19a19b (00 ) (00 d) d d or 00(00 d) d 00(19a d) d 00(19b ) d This shows that to alulate 19a 19b, we may do Step 1: Calulate 00 19a and d 0019b Step : Calulate 19 a d or 19 b Step 3: Calulate (19a d) or (19b ) Step 4: Calulate Step 5: Attah d d as TWO digits to the right of (19a d) or (19b ) Mental Calulation: n9a n9b The similar proedure applies to the multipliations in the form n9a n9b where n is a digit greater than Instead of we have to multiply n 1 in step 3 Example 5 Calulate n9 a d or n9 b by Step 1: Calulate and Step : Calulate or Step 3: Calulate Step 4: Calulate Step 5: Attah 1 to the right of 879: 8791 Then Example 6 Calulate Step 1: Calulate and Step : Calulate or Step 3: Calulate Step 4: Calulate Step 5 Attah 84 to the right of 154: We have Example 7 Calulate Step 1: Calulate and Step : Calulate or Step 3: Calulate Step 4: Calulate Step 5: Add 1 to 3444 yielding 3445, and attah 68 to the right of 3445: So Pratie Problems I Pratie Problems II Pratie Problems III Math Competition Skill Systematially Listing Aording to Shapes We have pratied systematially listing aording to shapes in ounting parallelograms in triangular grids (Issue 10, Volume 1) and ounting retangles in tableaus (Issue 16, Volume 1) This short lesson will present more examples Examples Example 1 (MathCounts State Sprint 1993 Problem 6) How many triangles of all sizes an you ount in the figure below? Copyright 009 Shuli Song shuli_song@yahooom All Rights Reserved Use with Permission

3 Shuli s Math Problem Solving Column Volume 1, Issue 18 April 16, 009 Answer: 35 Solution: We an lassify the triangles into six types Type One: We an ategorize the squares into five types Type One: There are 9 squares Type Two: There are 5 triangles Type Two: There are 5 triangles Type Three: There are 4 squares Type Three: or There are 10 triangles Type Four: There are 5 triangles Type Five: There are 5 triangles Type Six: There is only one square Type Four: There are 1 squares Type Five: There are 5 squares Altogether, the number of squares is There are 5 triangles Altogether, the number of triangles is Example How many squares of all sizes an you ount in the figure? Example 3 ( MathCounts Handbook Work-Out 8 Problem 8) On this 5 by 5 grid of dots, one square is shown in the diagram Inluding this square, how many squares of different sizes an be ounted using four dots of this array as verties? Answer: 31 Solution: Answer: 50 Solution: Copyright 009 Shuli Song shuli_song@yahooom All Rights Reserved Use with Permission 3

4 Shuli s Math Problem Solving Column Volume 1, Issue 18 April 16, 009 There are 8 types of squares For eah type we an obtain the number of squares using the Moving the Shape method Obviously there are squares of 1 1 The following figure shows the only one 8 8 square We plae a square at the left-top orner There are three positions inluding the urrent position to move it to the right and three positions to move it to the bottom So the number of squares is There are also two orientations of squares, whih are shown below There is only one in eah orientation Similarly we an obtain the number of 3 3 squares, whih is 4 So there are two squares Therefore, the total number of squares is Example How many regular hexagons are there whose verties are among the points of the following triangular grid? There is only one 4 4 square We plae a square at the left-top orner Using the Moving the Shape method we know that the number of squares in this type is Answer: 1 Solution: There are three types of regular hexagons Type 1: Side length 1 There are two orientations of 5 5 squares, whih are shown below For eah orientationthere are 4 squares Altogether, there are 4 8 squares of this type With the Moving the Shape method, we an find 15 regular hexagons of this type Copyright 009 Shuli Song shuli_song@yahooom All Rights Reserved Use with Permission 4

5 Shuli s Math Problem Solving Column Volume 1, Issue 18 April 16, 009 Type : Side length 3 Sixteen points are equally spaed as shown How many sets of four points are the verties of a square? 4 How many regular hexagons are there whose verties are among the points of the following triangular grid? We an ount 3 regular hexagons of this type by Moving the Shape Type 3: Side length Pratie Problems II We an also find 3 regular hexagons of this type with the same way Altogether, there are regular hexagons (All problems are from MathCounts) 1 (1990 MathCounts National Team Problem 1) How many squares are ontained in the figure? Pratie Problems I 1 How many triangles of all sizes an you ount in the figure? (1986 MathCounts State Individual Problem ) How many triangles of any size are ontained in the figure shown? How many squares of all sizes an you ount in the figure? 3 (001 MathCounts State Team Problem 10) How many equilateral triangles an be formed within the same plane using at least two verties that are also verties of a given regular hexagon? Copyright 009 Shuli Song shuli_song@yahooom All Rights Reserved Use with Permission 5

6 Shuli s Math Problem Solving Column Volume 1, Issue 18 April 16, (000 MathCounts Chapter Sprint Problem 14) How many different squares an be formed by using four of the evenly-spaed dots blow as verties of the square? (UNCMC Final Round Problem 7) Two points are randomly and simultaneously seleted from the 7 by 13 grid of lattie points m, n :1 m 13 and1 n 7 Determine the probability that the distane between the two points is an integer 5 (000 MathCounts State Sprint Problem 13) How many triangles are in the figure? 193 Answer: (1999 MathCounts National Sprint Problem ) How many squares are pitured? 7 (005 MathCounts Chapter Team Problem 4) How many triangles are in the figure? 8 (001 MathCounts State Sprint Problem 8) Sixty-four unit ubes are plaed together to reate a large ube How many ubes with integer dimensions are in the 4 44 ube? Solution: Systematially listing works well in this problem Note that There are ways to hoose two points If we hoose two points in a vertial line, these two points have an integral distane 7 There are ways to hoose two points in vertial lines If we hoose two points in a horizontal line, these two points have an integral distane 13 There are ways to hoose two points in horizontal lines If two points as the opposite verties make a 3 4 retangle, the distane between these two points will be 5 Now we have to ount the number of 3 4 retangles whose verties are in these points Remember the Moving the Shape method Plae a 3 by 4 retangle at the left-top orner There are 10 positions inluding the urrent position to move it to the right and 3 positions to move it to the bottom So there are retangles of 3 4, whose verties are in these points 5 A Problem from a Real Math Competition Today s problem omes from University of Northern Colorado Mathematis Contest (UNCMC) Copyright 009 Shuli Song shuli_song@yahooom All Rights Reserved Use with Permission 6

7 Shuli s Math Problem Solving Column Volume 1, Issue 18 April 16, 009 We also need to ount the number of 4 3 retangles whose verties are in the lattie points 5 Similarly there are retangles of 4 3 whose verties are in these points If two points as the opposite verties make a 6 8 retangle, the distane between the two points will be 10 We an ount 5 retangles of 6 8 At last we ount the number of 5 1 retangles, the length of whose diagonals is 13 There are retangles of 5 1 Altogether we have retangles with integral diagonals Every retangle has two diagonals There are diagonals, whih are integral Therefore, we have ways to hoose two points of an integral distane The probability is Pratie Problems II (UNCMC First Round Problem 10) Two points are randomly and simultaneously seleted from the 4 by 5 grid of 0 lattie points m, n :1 m 5 and 1 n 4 Determine the probability that the distane between the two points is an integer Pratie Problems II Pratie Problems III Systematially Listing Aording to Numbers Pratie Problems I Pratie Problems II A Problem from a Real Math Competition Note that and The answer is 44 The numbers of peanuts on the 44 dishes may be 1,,, 4, 43, and 54 respetively Solutions to Creative Thinking Problems 46 to Five Square to Four We have to make 4 squares with 16 mathstiks in the new shape We annot overlap any sides of any squares So we have to destroy the squares in the old shape, whih have the most sides overlapped with others Answers to All Pratie Problems in Last Issue Math Trik: Mental Calulation Pratie Problems I Here is the solution: Copyright 009 Shuli Song shuli_song@yahooom All Rights Reserved Use with Permission 7

8 Shuli s Math Problem Solving Column Volume 1, Issue 18 April 16, Digits from 1 to 9 From the addition, we have F 1, A 9, and 0 From the subtration, we see E 8 Now we have 1 0HI 8JD Note that I C 8 mod 10 and C 8 mod 10 So I D 6 mod10 Thus I D 6 or D I 4 D Sine 1, 0, 8, and 9 are not available, I D6 is impossible Then D I 4 So D 6, I or D 7, I 3 Case 1: D 6 and I Then C 4 Now we have H 8J6 In the addition, H That is, 3 H Sine only 3, 5, and 7 are available, it is impossible Case : D 7 and I 3 Then C 5 Now we have H3 In the addition, 110 So 6, H 4 or 4 48 Two Maps 9C + D8 9C D J7 7 H That is, H, H If 6 and H 4, then J The subtration annot be satisfied If 4 and H, then J 6 We have the solution: If the two maps are overlapped as shown below, the pin point is the left-top orner If the S map is at the enter of the L map, the pin point is the enter of the two maps If the sides of the two maps are parallel respetively, we an easily find the pin point In the following figure the two maps have the same orientation Draw the first line through the left-bottom orners of the two maps Draw the seond line through the left-top orners of the two maps Two lines interset at a point, at whih the same plae is represented in both maps This an readily be proved by similar triangles Now I remove the S map so that we an reognize the same plae in the L map If the two maps are overlapped as shown in the following figure, the pin point is the right-bottom orner Copyright 009 Shuli Song shuli_song@yahooom All Rights Reserved Use with Permission 8

9 Shuli s Math Problem Solving Column Volume 1, Issue 18 April 16, 009 In the figure below the sides of the two maps are parallel, but the S map is upside-down We an also find the pin point as shown a b d C Now we tear off the part of the L map outside retangle ACD A Again I remove the S map Now we an see the same plae in the L map D C We also tear off the orresponding part of the S map outside retangle abd a In general, the sides of the two maps are not parallel as shown below The following ontext is not a strit mathematial proof, but it is for you to believe that the fat is true In the figure below retangle ACD is the position of the S map on the L map A d Then we plae them together as their original C positions A a b b d D C C Now we have a smaller S map on a smaller L map These two smaller maps are the same exept their sizes Let retangle EFH be the position of the smaller S map on the smaller L map E D There is the orresponding retangle on the S map, whih is retangle abd C H F Copyright 009 Shuli Song shuli_song@yahooom All Rights Reserved Use with Permission 9

10 Shuli s Math Problem Solving Column Volume 1, Issue 18 April 16, 009 The orresponding retangle on the smaller S map is marked with efgh e Eventually, the two maps beome points eause we keep the S map and the L map always the same, the points represent the same plae on the two maps h f Clues to Creative Thinking Problems 49 to 51 Now we tear off the part of the smaller L map outside retangle EFH E H We also tear off the orresponding part of the smaller S map outside retangle efgh e h Then we put them together as their original positions again g g f F = 10 This is a triky question 50 Who Is Taller? Find somebody who an be ompared with TS and with ST 51 1 alls First weighing: 4 balls against 4 balls If the sale is in balane, the bad ball is in the third group of 4 balls Then it is not diffiult to determine the bad ball with two weighings If the sale is not in balane, assume that balls 1,, 3, and 4 are heavier than balls 5, 6, 7, and 8 Seond weighing: balls 1,, 5 against balls 3, 4, 6 Now go ahead Creative Thinking Problems 5 to 54 5 Make One Word Rearrange new door to make one word instead 53 Make 4 Equilateral Triangles With 3 mathstiks we an make one equilateral triangle With five we an make two equilateral triangles Now we have an even smaller S map on a smaller L map These two maps are the same exept their sizes If we tear off the outside parts of two maps again, the maps beome further smaller Make four equilateral triangles with six mathstiks without bending and breaking any mathstik If we do it again and again, 54 Another Challenge to Make 4 Make 4 with we will have a very small S map on a very small L map, whih are the same exept their sizes See the rules in Creative Thinking Problem 6 appearing in Issue, Volume 1 (Clues and solutions will be given in the next issues) Copyright 009 Shuli Song shuli_song@yahooom All Rights Reserved Use with Permission 10