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A Change in Code

Stage: 4 Challenge Level: Challenge Level:1

Good thinking from several people on this problem.
Below is a combination of solutions from Paul at Castleknock College, and Jean in Warwick

The answer to the first part is a 20% increase.

The most it can be is a 22.5% increase because 80 would then become 98, it can't be 99 or else there would be 2 equal letters.

All numbers have one thing in common -  they're divisible by 5

So the only way all of these numbers could produce integers, with no fraction or decimal part, is if their increase is calculated using fifths.

It must be 20% because anything greater, 40% 60% etc., would turn 80 into a three digit number which it clearly isn't.

An alternative explanation could be : 80 to HF is an increase but still only produces a 2 digit number, so HF is 81 to 99.
Trying each percentage, for example 80 to 90 is 12.5%, all give  results  involving fractions from some of the start values, except for 80 to 96 (20%).
And 20% checks as consistent across all data.

In Part 2 the answer is 25%
BC to 15 so B is 1 and C is 2, 3, or 4
CG to 35 so C is in fact 2 or 3  
C is not zero so BC is either 11, 12, or 13

11 to 15 and 13 to 15 do not give a percentage that keeps all other results integer.

12 to 15 is an increase of 25% and this is consistent for all the other data. 


In Part 3. the percentage increase was 40%

Increasing a number is like multiplying with a top-heavy fraction. For example a 5% increase is achieved multiplying the original number by 21/20. 
Dividing by 21/20 will reverse that process, and multiplying by 20/21 is the same thing.  

 I notice that all the results contain a factor of 7, so I am looking for a/7 as that multiplier, where a< 7, and choosing a so as to avoid fractions as required.
 EHJ goes to 147
 So EHJ is between 101 and 146, because it has 3 digits and this is an increase.
If EHJ/147 is equivalent to the fraction a/7 then EHJ is 105 or 126. Giving an a value of 5 or 6

Next I bring in CJH which increases to 1330

1330 multiplied by 6/7 is 1140, which is four digits not three, so 5/7 is the correct multiplier.

The percentage increase was 40%