Mathematics Magic Lam Pui Ki Wong Wai Ling

Mathematics Magic Lam Pui Ki Wong Wai Ling 1

Objective § Educational: § Use elementary mathematics to create geometric illusion, mathematics tricks § Learning mathematics is interesting! 2

Outline § Geometric vanishes § - Line paradox § - Vanishing face § - The leprechauns § The magic number, 9 § - Predicting the end point § - Trick about 3 digits number 3

Line Paradox 10 lines 4

Line Paradox (Activity) § Rules: § 1. cut the paper once only. § 2. shift the paper in a way which results in only nine lines. § Remarks: § All the lines should be of equal length and parallel to each other. 5

Line Paradox Cut into two parts 6

Line Paradox 9 lines 7

Where is the missing line? Above the diagonal: progressive decrease in the length Below the diagonal: progressive increase in the length 8 of the 10 lines are broken into 2 segments The 16 segments are redistributed to form 9 new longer lines The small increases exactly equal the length of the missing line 8

Vanishing Face Activity Add a line and shift the picture to make one of the faces disappear. 9

Vanishing Face 10

Math explanations Line Paradox δ is the small increase of the new line L 1 +δ L 2 L 3 L 4 …. . L 10 -δ -2δ -3δ +2δ +3δ +4δ -9δ _____________________________________ 11

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Math explanations Vanishing Face δ is the small increase of the area of the new face A 1 A 2 A 3 -δA 2 -δA 3 A 4 … A 6 -δA 4 -δA 6 +_____________________________________ δA 2 +δA 3 +δA 4 +δA 5 13

Leprechauns 14

Other example 15

Other example 16

Magic nine 17

Mystery of nine Activity Predicting the end point 18

Mystery of nine Rules 1. Choose a number > number of coins in the tail 2. Count from the end of the tail, counting upward and around the circle anticlockwise until reaching the chosen number 3. Start counting at 1, count clockwise around the circle 19

Mystery of nine Observation Counting ends at the same point ! 20

Trick of 3 digits number Step 1: think of any 3 digits number (e. g. abc) requirement: a≠c Step 2: reverse the order (abc→cba) Step 3: subtract the smaller 3 digits number from the larger one Step 4: add the result to its reverse ordered number. 21

Explanation observe the result in step 3: - middle digit = 9 - 1 st digit + 3 rd digit = 9 The result in step 3 is in the form: α 9β, where α+β = 9 Step 4: α 9β + β 9α = 1098 22

Online discussion 1. Give a daily life application of the geometric vanishes 2. Can you give other mathematics magic or tricks? 23

Homework 1. Cut the American flag of 13 stripes into the fewest number of pieces that can be fitted to form a flag of 15 stripes. [Just draw the line that you will cut on the flag] 24

Homework 2. Give another example of geometric vanishes and explain it. Extra credit Modify the example shown in slides 20 and 21. Show to make the result in step 3 becomes α 7β, where α+β = 7. 25