On 8/9/2012 1:31 PM, andrel wrote:
> On 9-8-2012 12:15, Invisible wrote:
>> Warp's challenge on prime numbers got me thinking...
>>
>> 5 + 5 = 10
>> 5 + 8 = 13
>> 5 + 5 + 5 = 15
>> 8 + 8 = 16
>> 5 + 5 + 8 = 18
>> 5 + 5 + 5 + 5 = 20
>> 5 + 8 + 8 = 21
>> 5 + 5 + 5 + 8 = 23
>> 8 + 8 + 8 = 24
>> ...
>>
>> In particular, there is no way of making 27, but you can make 28 just
>> fine (5+5+5+5+8). Actually,
>>
>> 28 = 5+5+5+5+8
>> 29 = 5+8+8+8
>> 30 = 5+5+5+5+5+5
>> 31 = 5+5+5+8+8
>> 32 = 8+8+8+8
>> 33 = 5+5+5+5+5+8
>> 34 = 5+5+8+8+8
>>
>> According to my simulations, you can make /every/ number above 27. So 27
>> is the highest number that you cannot make.
>>
>> When you get to 40, something interesting happens. For 5*8 = 40, which
>> means that we have
>>
>> 5*8 = 5+5+5+5+5+5+5+5 = 40
>> 8*5 = 8+8+8+8+8 = 40
>>
>> So this number can be made in /two/ different ways.
>>
>> When we come to 45, the same happens again, since 45 = 40+5, and 40
>> itself can be made two different ways. So you can take both of those
>> ways and append +5 to make 45 in two different ways.
>>
>> The same happens again with 48 = 40+8.
>>
>> It happens yet again with 50 = 40+10. 10 can only be made one way, but
>> 40 has two ways.
>>
>> It happens yet again with 13, since 53 = 40+13, and 13 is makable.
>>
>> In fact, above 27, /every/ number can be made at least one way, 40 is
>> the first number that can be made /two/ ways, and every number which can
>> be made by adding a makable number to 40 also has two ways.
>
> Does that make 67 the highest number you can make in only one way?

If you generalize to allow the factors of 8 and 5 to be negative 
integers, then every integer can be made an infinite number of different 
ways.

Regards,
John

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