ektalks: A Tutorial on Alphamatic Puzzles - Wonderful Brain Training to Enhance Cognitive Function

Since my retirement 20 years ago, I have spent a lot of time in brain training activities - they say that exercising the brain is important for keeping age-related cognitive problems at bay. I totally subscribe to this theory provided the activities are enjoyable, challenging and address different cognitive domains of the brain. Playing Scrabble will sharpen a few of the skills relating to vocabulary, spellings, memory, spatial awareness but can only be one of the many different games and puzzles that one needs to indulge in. I also like alphamatic puzzles as they help analytical reasoning and logic as well as boosting short term memory, attention and concentration - also they require knowledge of only school-level maths and everybody can enjoy them.

Alphamatic (aka cryptarithmetic) is a game where digits have been replaced by letters in an arithmetic operation - each letter represents a unique digit, with no two letters having the same value. The goal is to find the digits - 0 to 9 - that the letters represent such that the resulting arithmetic operation is true.

Slides A1 and A2 of the appendix explain the terms used in basic arithmetical operations with particular emphasis on digits carryover.

As a simple example, consider the following puzzle:

In this tutorial, I shall use the notation of numbering the columns from the right (rightmost column is Clm1 etc.) and identify the carry as cif where i is the number of column that generated it and f is the next column to the left of column i, or f = i + 1) (see appendix slide A1 and A2 for more details) - for example, the Units column is Clm1 and the digit carried over to the Tens column (Clm2) is c12. Similarly, the digit carried over from the Tens column (Clm2) to the Hundreds column (Clm3) is c23.

Also, by convention the number in the leftmost column is not a zero.

Carry plays an important role in solving the puzzle. It is also useful to look at the leftmost column as this can help to give a good idea of the range of values that the letters have in there. If the sum row contains a higher value column not present in other rows, then the letter in the leftmost column will have a value equal to the carry that may be 1 or 2 (generally the value of a carry is 0, 1 or 2 , and rarely 3) - see slide 1 where A = 1. If a higher value column is not present then in the leftmost column the sum of all the letters in rows + any carry from the previous column will be <10 - a useful piece of information.

Keeping track of all the details can get quite involved and one may lose direct awareness of the world outside - this might have a wonderful meditative effect that calms the system - particularly if you reach a successful result!

Let us look at another example to practice the above information:

Another example of a similar puzzle is as follows:

If the number ABCDEF is multiplied by 3 then it becomes BCDEFA. Can you find the number?

The solution is given in appendix slide A3 - but try to solve the puzzle yourself first. Interestingly, the puzzle has two solutions showing two different numbers have this property. Both answers are given in slide A3.

A puzzle that is slightly more difficult is described in slides 4 and 5. Study the analysis to understand how to approach the solution. Again, there is no unique way of solving a puzzle and you might wish to try your own method.