uthors compare mathematical induction to dominoes toppling in succession. (Gunderson, 2011, p. 4). Suppose that: 1) We can knock down the first domino, 2) the dominos are so close, that each previous will knock the following one down when falling. Then all the dominos will be down, as shown in Figure 2. p align="justify"> Another analogy for mathematical induction is given by Hugo Steinhaus in Mathematical Snapshots in the 1983 (Steinhaus, 1983, p. 299). Consider a pile of envelopes, as high as one likes. Suppose that each envelope except the bottom one contains the same message "open the next envelope on the pile and follow the instructions contained therein." If someone opens the first (top) envelope, reads the message, and follows its instructions, then that person is compelled to open envelope number two of the pile. If the person decides to follow each instruction, that person then opens all the envelopes in the pile. The last envelope might contain a message "Done". This is the principle of mathematical induction applied to a finite set, perhaps called "finite induction". Of course, if the pile is infinite and each envelope is numbered with consecutive positive integers, anyone following the instructions would (if there were enough time) open all of them; such a situation is analogous to mathematical induction as it is most often used. understand the method of mathematical induction, several teachers of mathematics both in Latvia and abroad, make students solve the task about the Towers of Hanoi, invented by the French mathematician Edouard Lucas in 1883. Task 1: three rods and a number of disks of different sizes are given. Only smaller disks may be placed on larger disks. All disks from the first rod have to be moved to the third rod by employing minimum moves, as shown in Figure 3. Several mathematicians have invented programs for visual solution of this task. For example, Figure 4 shows that applet is based on the Tower of Hanoi. Applet created by David Herzog (Pierce, 2008). br/>В
Figure 3. Tower of Hanoi
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Figure 4. Interactive solution of the task
Many teachers ask their students to create visual models in order to understand mathematical induction. For example, Task 2: At a party, everybody shakes hands with all attendees. If there are n people at the party and each person shakes the hand of each other person exactly once, how many handshakes take place? Handshakes may be described visually, where persons are marked as circles, but handshakes as line segments, as shown in Figure 5:
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Figure 5. Visual interpretation of Task 2
The figure demonstrates that the number of handshakes for one person equals to 0, two persons have one handshake, three persons - 3 handshakes, four persons - 6 handshakes, five persons - 10 handshakes and six persons - 15 handshakes. Students can further make t...
