Watch what happens when we take the numberĀ 192 and first double it, then triple it:

192 192 x 2 x 3 384 576

Now we have three numbers: 192, 384, and 576. Do you see anything strange about them?

Now watch again. If we move the 2 from the back of 192 to the front, we will have 219.

Try the double-triple process once more.

Does the same strange thing happen this time?

It will, if you are careful.

There are two more 3-place numbers that will work for our double-triple game. Can you find at least one of them?

Here is a hint: The sum of the digits of these new numbers is always 12.

(Note: this was also true for 192: 1 + 9 + 2 = 12.) It’s not as hard as it looks.

After you have found one of the new numbers that I just asked you to find, please fill out the table below and tell me about any interesting patterns that you notice.

First case | Sum of Digits | New Number | Sum of Digits | |

number | 192 | x | x | x |

double | 384 | x | x | x |

triple | 576 | x | x | x |

Source for idea: H. E. Dudeney, Amusements in Mathematics. Dover, 1958. pp. 14 & 155.

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## Update: August 2003

The chart above is meant to show an important numerical relationship between the original number and its double and triple. Here is another chart that can be used to expand that concept.

Find the sums of the digits in each row, column, and main diagonal, much in the same way as is done for magic squares. Place those sums in the corresponding circles around the square. Examine those sums carefully. You should see an important property that connects those numbers.Then you could re-draw this figure and this time place the digital roots in the circles, to see yet another pattern. |