# Distinct digits but Close Enough

Using the digits from $1$ to $9$,

• Form four numbers, such as ($X$,$Y$,$Z$,$T$)
• Then take $X\div Y$ and $Z\div T$ where both values should be different than each other.
• You need to use all digits from $1$ to $9$ but once, such as ($923\div 85$,$67\div 41$)

What is the smallest difference between these two values?

• how many digits for each number, any restrictions? – Shahriar Mahmud Sajid Sep 17 '18 at 11:24
• Strange you accepted the answer that provided solution later. – rus9384 Sep 17 '18 at 19:05
• @rus9384 what do you mean by solution later? – Oray Sep 17 '18 at 19:05
• See when the edits were commited. Just hover your mouse over "edited x hours ago" link. – rus9384 Sep 17 '18 at 19:06
• @rus9384 seems you are right, I missed the other one who provided the right answer before anyone else :) thanks – Oray Sep 17 '18 at 19:08

My first attempt

Staying close to the original order of numbers
45/69 - 78/123
0.01802757

No major breakthrough, just another attempt

Keeping the top low and using ad odd number of total top digits
12/987 - 5/436
0.000690164803

Third

2/835 - 4/1679
0.00001283912223

This one is pretty small

Again keeping the top tiny and trying to keep the bottom factor as close to the factor on top
1 / 826 - 9 / 7435
0.000000162831708
(7435 / 826 = 9.00121)

• I wrote a computer program, and your last solution seems to be the best possible. – Jaap Scherphuis Sep 17 '18 at 13:02
• @JaapScherphuis yes, I did the same and it doesn't get lower than that – NoOorZ24 Sep 17 '18 at 13:13
• As this one got smallest possible answer 1st I think this should be accepted – NoOorZ24 Sep 17 '18 at 13:14
• I feel I got a bit lucky here, as @Bass inspired my approach and the factor chosen was a bit arbitrary. Anyway, fun puzzle. It made quite a few people push for perfection, and the improvement of answers from different people was fun to see! – P1storius Sep 17 '18 at 13:19
• By the way, I took a look at all the non-answers where the difference is zero. These range from $1/8 - 954/7632=0$ where there is a factor of $954$ between their denominators or numerators, to $32/96 - 58/174=0$ where the factor is just below $2$. – Jaap Scherphuis Sep 17 '18 at 14:57

First attempt:

Make two rational numbers, with one's numbers twice that of the others, small numbers on top, large numbers on the bottom: $\frac{4}{763} - \frac{8}{1529} \approx .000010$ (four zeroes after the decimal point)

Second attempt:

Factor of three: $\frac{1}{954} - \frac{3}{2867} \approx .0000018$ (five zeroes)

Third:

$\frac{4}{3567} - \frac{1}{892} \approx .000000314$ (six zeroes)

Fourth:

$\frac{1}{826} - \frac{9}{7435} \approx .00000016$ (still six zeroes)

I think it might be

0.

This is because

$\frac{1358}{4} - \frac{679}{2} = 0$.

• "both values should be different than each other." – Oray Sep 17 '18 at 12:38
• given that the values must be different, 0 seems unlikely. – Bass Sep 17 '18 at 12:38
• Sorry gentlemen, I just can’t read! – El-Guest Sep 17 '18 at 12:38
• though still thumps up, good one :) – Oray Sep 17 '18 at 12:46

$$\frac{2}{3}-\frac{987654}{1} = -987653.333\ldots$$ This is much smaller than your near zero answers.

• Hahahahahahaha. Hilarious! Good one! Although technically a whole number is a fraction, and he didn't say we can't do exponents, so, 1 - (9^8^7^6^5^4^3^2) is way better. – Alto Sep 17 '18 at 17:18

Just by randomly mixing up different digits I came up with this

$\dfrac{64}{8} - \dfrac{975}{123} = 0.0731707\dots$

Second try

$\dfrac{124}{975} - \dfrac{8}{63} = 0.000195\dots$

• You can do much better by using the reciprocal values. – M Oehm Sep 17 '18 at 11:56
• @MOehm oh god, thats true, feel stupid now – Ian Fako Sep 17 '18 at 11:58

Here's My attempt-

$\dfrac{975}{483} - \dfrac{12}{6} = 0.01863354\dots$