1. A RITHMETIC PRESENTATION By: Alexandra Silva & Dani Hoover

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

3. T2T2 Example: Start with the number 49 94-49=45 54-45=09 90-09=81 81-18=63 63-36=27 72-27=45 54-45=09 Pattern starts to repeat! Example 2: Start with the number 24 42-24=18 81-18=63 63-36=27 72-27=45

4. T3T3 Example: Start with the number 123 321-123=197 971-179=792 972-279=693 963-369=594 954-459=495 *Therefore, we conclude that all T 3 (with the exception of aaa) converge to a fixed point 495.

5. T4T4 Example: Start with the number 2143 4321-1234=3087 8730-0378=8352 8532-2358=6174 7641-1467=6174 *Therefore, we conclude that all T 4 (with the exception of aaaa) converge to a fixed point 6174.

6. T5T5 Example: Start with the number 54321 54321-12345=41976 82962 75933 63954 61974 82962

7. C ONCLUSIONS FOR T 1 THROUGH T 10 T 1 – Can’t do. T 2 – repeated pattern; starts at 45 T 3 – converges to fixed point 495 T 4 – converges to fixed point 6174 T 5 – repeated pattern; starts at 82962 T 6 – repeated pattern; starts at 851742 T 7 – repeated pattern; starts at 8429652 T 8 – repeated pattern; starts at 75308643 T 9 – repeated pattern; starts at 863098632 T 10 – repeated pattern; starts at 8633086632

8. G ENERALIZATIONS Sum of the digits at the fixed point or at which the pattern starts to repeat follows another pattern: T 2 : 45: 4+5=9 T 3 : 495: 4+9+5=18 T 4 : 6174: 6+1+7+4=18 T 5 : 82962: 8+2+9+6+2=27 T 6 : 851742: ….…………..=27 T 7 : 8429652:….………….=36 T 8 : 75308643:….………...=36 T 9 : 863098632:…….…….=45 T 10 : 8633086632:…..…….=45

9. R ECAP Tn converges to a fixed point for T 3 and T 4. For all other T n (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 T n, we can predict the sum of the digits using the following equations: When n is even: sum T n = 9 (n/2) When n is odd: sum T n = 9 ((n+1)/2) (applies for n >1)