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Multiplication Magic

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.

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Transposition Fix

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine detect transposition errors in these numbers?

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Check Codes

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check if they are valid identification numbers?

Odd Stones

Age 14 to 16 Challenge Level:

$27$ stones are distributed between $3$ circles

On a "move" a stone is removed from two of the circles and placed in the third circle.

So, in the illustration, if a stone is removed from the $4$ and the $10$ circles and added to the $13$ circle, the new distribution would be $3$ - $9$ - $15$

Check you can turn $2$ - $8$ - $17$ into $3$ - $9$ - $15$ in two "moves"

Here are five of the ways that $27$ stones could be distributed between the three circles :

$6$ - $9$ - $12$

$3$ - $9$ - $15$

$4$ - $10$ - $13$

$4$ - $9$ - $14$

$2$ - $8$ - $17$

There is always some sequence of "moves" that will turn each distribution into any of the others - apart from one.

Identify the distribution that does not belong with the other four.

Can you be certain that this is actually impossible rather than just hard and so far unsuccessful?