## Teaser 3031: End of the beginning

**From The Sunday Times, 25th October 2020** [link]

Jenny is using her calculator, which accepts the input of numbers of up to ten digits in length, to prepare her lesson plan on large numbers. She can’t understand why the results being shown are smaller than she expected until she realizes that she has entered a number incorrectly.

She has entered the number with its first digit being incorrectly entered as its last digit. The number has been entered with its second digit first, its third digit second etc. and what should have been the first digit entered last. The number she actually entered into her calculator was 25/43rds of what it should have been.

What is the correct number?

[teaser3031]

## Jim Randell 4:59 pm

on23 October 2020 Permalink |See also:

Enigma 1036,Enigma 1161,Teaser 2565.If we have a number, where the leading digit is

aand the remainingkdigits arebcdefg… = r, then we have:The following Python program runs in 45ms.

Run:[ @replit ]Solution:The correct number is: 530864197.So the number entered is: 308641975.

For this particular puzzle we can do some analysis can reduce the solution space further. (From the 9×9 = 81 cases to consider in the general case).

In the equation:

we see that that each side is divisible by 5, so

amust be 5 (as it cannot be 0).Which leaves:

Which we can then check with the 9 possible values for

kmanually or with an even shorter program:Or, for a complete manual solution:

Numbers of the form:

(25.10^k − 43) / 9, look like this:To divide by 9 again we need the number to have a digital root of 9, and the only one in the required range is:

Dividing this by 9 gives:

Hence the correct number is 530864197, and the incorrect number 308641975.

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## hans droog 10:01 am

on6 November 2020 Permalink |Hi Jim. would be obliged if you could explain formula in teaser 3031. Many thanks, Hans Droog

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## Jim Randell 10:10 am

on6 November 2020 Permalink |@hans:

As an example, if we had an 7 digit number

abcdefgwhich was entered incorrectly on the calculator asbcdefga, then we would have:If we represent the 6 digit number

bcdefgasrwe can write this expression as:The expression I give in my solution is the general case when

ris akdigit number.(“.” is multiplication. “^” is exponentiation).

Is that clear?

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## Frits 1:42 pm

on24 October 2020 Permalink |Similar.

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## Hugh Casement 4:15 pm

on7 November 2020 Permalink |Hans may have been put off by Jim’s use of a decimal point to mean multiplication.

The international convention is a raised dot.

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## Jim Randell 10:13 am

on8 November 2020 Permalink |If I were handwriting a decimal number I would use a “middle dot” for the decimal point, and a dot on the line to denote multiplication. Of course when typing the “.” (full stop) symbol has to do for both.

Fortunately we don’t often have to deal with numbers with decimal points in puzzles.

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## hans droog 9:01 am

on8 November 2020 Permalink |thats right Hugh

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