Reducing 1M to 7: Is There a Shorter Way?

[recon]

Can you reduce 1,000,000 to 7 using only the functions +,-,X and / with the digit 7?

I know a lengthy way of doing it - $$[1,000,000 - \frac{7}{7}] \div 7 \div 7 \div 7 ...$$.

Is there a shorter way?

 

[The Bob]

I know a lengthy way of doing it - $$[1,000,000 - \frac{7}{7}] \div 7 \div 7 \div 7 ...$$.

$$(1,000,000 - \frac{7}{7}) \div(7^7) \div(7^7)...$$

It is a bit shorter.

The Bob (2004 ©)

EDIT: $$(1,000,000 - \frac{7}{7}) \div(7^7) = 1.2142645$$ That decimal run does not seem to have a fraction that can be put to it so I am stuck on how you got 7 from what you had.

$$[(1,000,000 - \frac{7}{7}) \div(7^7)]+[(7\times\frac{7}{7})-\frac{7}{7}]= 7.2142645$$ which is close.

 

[ceptimus]

$$\left(\frac{77 - 7}{7}\right)^{(7 - \frac{7}{7})}$$

 

[recon]

I forgot to mention that you're supposed to solve this by using a basic calculator that has no 'power' function. This renders the two solutions above to the realm of the 'lengthy'.

 

[ceptimus]

$$\left(\frac{77 - 7}{7}\right)\left(\frac{77 - 7}{7}\right)\left(\frac{77 - 7}{7}\right)\left(\frac{77 - 7}{7}\right)\left(\frac{77 - 7}{7}\right)\left(\frac{77 - 7}{7}\right)$$

or

$$\left(777 + 7 \times 77 - \left(7 + 7 + 7 + 7 + \frac{7}{7}\right)\right) \times 777 + \frac{7}{7}$$

or

$$\frac{7,777,777 - 777,777}{7}$$

or even

$$\frac{777777}{.777777}$$

 

[recon]

Sorry ceptimus, but you've got to start out with the number 1,000,000 and then reduce this number to 7 using anyone of the basic functions mentioned in my earlier post.

 

[ceptimus]

$$1,000,000 \times \frac{.777777 \times 7}{777777} = 7$$

 

[recon]

Sorry to disappoint you again. It seems that I have not explained the problem properly. Try to imagine that you're using a calculator that has only the buttons $$1,0,7,+,-,\div, \times, =$$. You may only use the 1 and 0 buttons for inputing the number 1,000,000 and for nothing else.

So you can now see that it is not possible to use decimal points the way you used it. :)

 

[ceptimus]

How about

$$1,000,000 \times (7 - 7) + 7 = 7$$



OK, that still uses brackets, and it's cheating.

 

[recon]

I like that! However, I think that's cheating...

Then again, I also did cheat with the brackets in my first post.

 

[ceptimus]

This works on a basic 4-function calculator without a decimal point.

$$1,000,000\;-\;7\;\div\;7\;=\;\div\;777\;-\;7\;\times\;77\;+\;7\;+\;7\;+\;7\;+\;7\;+\;7\;\div\;7\;=\times\;7\;\div\;777\;=$$

 

[Gokul43201]

Can you reduce 1,000,000 to 7 using only the functions +,-,X and / with the digit 7?

I know a lengthy way of doing it - $$[1,000,000 - \frac{7}{7}] \div 7 \div 7 \div 7 ...$$.

Did you mean :
$$([1,000,000 - \frac{7}{7}] \div 7) - 7 - 7 ... ~~~?$$

 

[Gokul43201]

If you're allowed to hit the 1/X button (which you're not), there's an easy way :

(1,000,000 + (777,777/7) )/7,777,777...and take the reciprocal.

 

[recon]

Did you mean :
$$([1,000,000 - \frac{7}{7}] \div 7) - 7 - 7 ... ~~~?$$

Actually, yes. But it still won't work.

 

[Bartholomew]

How about

$$1,000,000 \times (7 - 7) + 7 = 7$$



OK, that still uses brackets, and it's cheating.

If you don't have to say 1,000,000 _first_, (7-7)*1000000 + 7 can be done on a basic calculator.

 

[Bartholomew]

Or, if you take advantage of the limited # of digits that can be stored in a basic calculator, you can do something like 1000000 / 7777777777777777 + 7

 

[recon]

I think this may be the hardest brain teaser, yet.

It looks so simple, though...

 

[meL]

1000000 1001 /

 

[LarrrSDonald]

Very interesting question.

1000000 - 777777 - 77777 (repeat twice) - 7777 (repeat 8 times) - 777 (five times) - 77 (seven times) - 7 (four times) *7

would be one rather convoluted solution. I'm sure there's something better though, some number up there ought to be an even multiple of 7, 77, 777, etc. I'm seeing if I can find a more solid solution.

 

[LarrrSDonald]

Wow, I exahausive searched 7k, 77k, 777k.. with each other over +-10 and not a single number is divisible by 7 (or 77, 777 etc, though that would have been kind of a bounus). Unless I did it wrong (and it's entirely possible that I did) I can't find anything. There are several +- solutions that end up with 1 like the one I did above, like:
1000000
-777777
-77777 (3x)
+7777
+777 (4x)
+77 (3x)
-7
*7
for instance. I'm not sure what's considered the "shortest", like fewest buttons pushed? I might do a more solid optimize if I have time or if someone else would enjoy it then go ahead.

[EDIT] Doh.. Adding various multiples of 7 won't have much of an effect on divisibillity by 7, will it now? I'm still thinking about the posibillity of slight mods like *7+7/7 (i.e. +1) in strategic places though, that's probably the ticket if there is a trivial solution.

 

[LarrrSDonald]

Been fiddling around a bit, here's one
1000000
*7
-7
/7
/777
-777
+7
+7
+7
+7
*7
+7
/7
/77

20 7s, 33 keystrokes not counting the million. It could probably be polished down a bit I'm sure. The essential trick here is that one can use *7 (several +-7, 77, 777) /7 to accomplish arbitrary additions (by 1, 11, 111 ...) to set things up for divides. There may be a need for = here and there, I don't have really have a "normal" calculator around to test it on.

 

[Mk]

Look at all the cool people that can do it in LaTeX...

 

[Joffe]

1 000 000
* 7
- 7
/ 777 777
- 7
* 7
- 7

11 sevens, 18 keystrokes.