Arithmetic presentation By: Alexandra Silva & Dani Hoover

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?

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!

T3 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 T3 (with the exception of aaa) converge to a fixed point 495.

T4 Example: Start with the number 2143 4321-1234=3087 8730-0378=8352 8532-2358=6174 7641-1467=6174 *Therefore, we conclude that all T4 (with the exception of aaaa) converge to a fixed point 6174.

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

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 851742 T7– repeated pattern; starts at 8429652 T8– repeated pattern; starts at 75308643 T9– repeated pattern; starts at 863098632 T10– repeated pattern; starts at 8633086632

Generalizations Sum of the digits at the fixed point or at which the pattern starts to repeat follows another pattern: T2: 45: 4+5=9 T3: 495: 4+9+5=18 T4: 6174: 6+1+7+4=18 T5: 82962: 8+2+9+6+2=27 T6: 851742: ….…………..=27 T7: 8429652:….………….=36 T8: 75308643:….………...=36 T9: 863098632:…….…….=45 T10: 8633086632:…..…….=45

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)