1. Arithmetic presentation
By: Alexandra Silva & Dani Hoover

2. Goals: Does Tn converge to a fixed point for all n?
Does it always take the same number of steps?
Can we make any generalizations for Tn?

3. T2 Example: Example 2: Start with the number 49
94-49=45
54-45=09
90-09=81
81-18=63
63-36=27
72-27=45
Example 2:
Start with the number 24
42-24=18
81-18=63
63-36=27
72-27=45
Pattern starts to repeat!

4. T3 Example: Start with the number 123 321-123=197 971-179=792
=693
=594
=495
*Therefore, we conclude that all T3 (with the exception of aaa) converge to a fixed point 495.

5. T4 Example: Start with the number 2143 4321-1234=3087 8730-0378=8352
=6174
=6174
*Therefore, we conclude that all T4 (with the exception of aaaa) converge to a fixed point 6174.

6. T5 Example: Start with the number 54321 54321-12345=41976 82962 75933
63954
61974

7. Conclusions for t1 through t10
T1– Can’t do.
T2– repeated pattern; starts at 45
T3– converges to fixed point 495
T4– converges to fixed point 6174
T5– repeated pattern; starts at 82962
T6– repeated pattern; starts at
T7– repeated pattern; starts at
T8– repeated pattern; starts at
T9– repeated pattern; starts at
T10– repeated pattern; starts at

8. Generalizations
Sum of the digits at the fixed point or at which the pattern starts to repeat follows another pattern:
T2: 45: =9
T3: 495: =18
T4: 6174: =18
T5: 82962: =27
T6: : ….…………..=27
T7: :….………….=36
T8: :….………...=36
T9: :…….…….=45
T10: :…..…….=45

9. Recap
Tn converges to a fixed point for T3 and T4. For all other Tn (2 through 10) they have repeated patterns.
For different values, the number of steps to reach the fixed point or the repeated pattern varies.
For Tn, we can predict the sum of the digits using the following equations:
When n is even: sum Tn= 9 (n/2)
When n is odd: sum Tn= 9 ((n+1)/2)
(applies for n >1)