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May 29, 2015 · Given the sequence 123456789: You can insert three operations (+ +, − −, × ×, / /) into this sequence to make the equation = 100. My question is: is there a way to solve this without brute force? (I tried to represent it as a graph but I'm unsure where to go from there.) With brute force: 123 − 45 − 67 + 89 = 100 123 − 45 − 67 ...
During the computations the order of $123456789$ cannot be broken. For example $$1+2+3+4+5+6+7+(8×9)=100$$ They have given only this example. I have found that $$1×(-2+3+4)+(5×6)+(7×8)+9=100$$ $$(1×2×3)-(4×5)+(6×7)+(8×9)=100$$ It is easy to prove that number of ways of constructing $100$ is finite. But find all possible constructing ...
Dec 31, 2019 · $\begingroup$ Some questions which seem a bit related: How can I prove this odd property?, Why is $\frac{987654321}{123456789}$ almost exactly $8$?, Why is $\frac{987654321}{123456789} = 8.0000000729?!$ and maybe some of the questions linked there. $\endgroup$ –
May 20, 2013 · 19. 98765432 / 12345679 = 8, exactly. You can see how the pattern works by multiplying out 12345679 ∗ 8 starting at either end. This explains why your fraction is close to an integer. If you think the 729 is interesting (I don't), it can be explained by some of the other answers here.
Jan 11, 2013 · It's not entirely coincidence: 246,913,578 = 2·123,456,789. So of course that's what you get if you multiply by 5 or divide by 2. But I find it surprising that that's what you get if you multiply by 7, and I think there's something else at work here. It's also suspicious that none of those numbers are divisible by 3.
Jan 20, 2017 · This pattern is due to the fact that $9$ is one less than the basis of the numeration so that the products with individual digits (from the right $81,72,63,54\cdots$) have an increasing unit digit, while the tens digit increases. Thus thanks to carries, the digits of the product remains constant (with an anomaly).
Mar 24, 2018 · Then the results of B and C is multiplied and the result is 123456789. What number did A choose? For clarity: Let the number chosen by A be a, the number chosen by B be b, and the number chosen by C be c. We then have: $$(b^2 + a)(c^2\cdot a)= 123456789.$$ How would you solve this problem without a calculator?
In the first 1 billion digits of $\pi$, I found two instances of $123456789$, but no instances of $1234567890$. Here's a simple example. In the first billion digits, there were $10049$ instances of $12345.$ There were $969$ instances of $123456$. There were $97$ instances of $1234567$. There were $9$ instances of $12345678$. And there were two ...
Oct 4, 2016 · Related to this question, What is the smallest prime number made of sequential number? are there infinitely many primes of the following form (OEIS A057137)? $1, 12, 123, 1234, 12345, 123456, 12...
May 7, 2017 · I just started typing some numbers in my calculator and accidentally realized that $\frac{123456789}{987654321}=1/8$ and vice versa $\frac{987654321}{123456789}=8.000000072900001$, so very close to $8$. Is this just a coincidence or is there a pattern behind this or another explanation? I tried it with smaller subsets of the numbers but I never ...
