In last Monday’s puzzle, we asked you to figure out the next number in three different sequences, and many Lab readers did so brilliantly. Now, as promised (and courtesy once again of our Quiz Whiz Pradeep Mutalik), we have some more challenges concerning those same sequences. (If you didn’t solve last week’s yet, it’s never too late to try.)

To refresh your memory, here are the first two sequences from last week:

A) 8, 16, 22, 26, 38, 62 . . .

B) 8, 16, 23, 28, 38, 49 . . .

Now, without actually extending these sequences, predict whether sequence A or sequence B will have the larger number after adding another 10 terms. Why?

In sequence B, find a formula for the sum of the digits in all the numbers in the sequence (no matter where you stop), given the first and the last number (where you stopped) only.

In sequence B, each number generates the next number. Can you find the smallest number that can be generated by more than one number?

Now recall the remaining sequence presented last week (in which each number is followed by a pair of clues in parentheses to help find the next number):

C) 8127 (8721, 1278), 7443 (7443, 3447), 3996 (9963, 3699), 6264 (6642, 2466) . . .

This week’s final challenge is to extend sequence C until two consecutive numbers are the same (it won’t take very long). Then try generating sequences following the sequence C rule, starting with any four digit number of your own choice such that all four digits are not the same. Extend the sequence until two consecutive numbers are the same (it won’t take very long). Try this for three or four times. What do you find? Can you explain why this happens no matter where you start?

As usual, you can post answers as comments (and I won’t publish any of the comments until tomorrow). The weekly prize, a copy of “Mathematical Puzzles of Sam Loyd” (edited by Martin Gardner), will go to someone who comes up with an especially interesting answer (like one in verse) or to someone who proposes a sufficiently intriguing puzzle for Lab readers to solve.

To submit a puzzle, send an email message with “NEW LAB PUZZLE” in the subject line to tierneylab@nytimes.com. Please include the solution to the puzzle and indicate whether or not the puzzle is original.

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