## Find max flow in a network with all capcities of sqrt(2) and one with 2

Given a graph $$G(V, E)$$ with capacities on the edges such that all edges have a capacity of $$sqrt(2)$$ apart from one edge with a capacity of 2. need to find max flow efficiently.

I can run Dinic on this graph or FordFulkerson but I know it can be more time-efficient.

What I have tried –

transform all edges capacity to be 1, then find max flow using Dinic algorithm on 0-1 network that is more efficient than a general network. then If the edge that had a capacity of 2 isn’t saturated it won’t be saturated on the original graph so we can just find the min-cut and multiply the number of edges that cross the min-cut by sqrt(2) and that’s the max flow. but if it’s saturated I’m stuck.

## Can we represent $\sqrt{2}$ exactly even with infinite bits in mantissa

Can we represent $$\sqrt{2}$$ exactly even with infinite bits in mantissa in floating point notation or otherwise. We actually have to prove this is not possible. But why can’t we if we have infinite bits? Infinite bits means the binary representation of the mantissa can be indefinitely large. The mantissa is in IEEE 754 32 bit format and in a hypothetical world where we don’t have a bound on the number of bits which can be used to represent it.

## Are the digits of $\log 2$ and $\log 3$, or $\sqrt{2}$ and $\sqrt{3}$, or $e$ and $\pi$, cross-correlated?

I try to find sequences of digits (in base $$b$$, with $$b$$ not necessarily an integer) that are not cross-correlated. While the digits in base $$b$$ of (say) $$\pi$$ and $$e$$ do not exhibit cross-correlations when taken separately (assuming $$b$$ is an integer) since these numbers are believed to be normal numbers, what about cross-correlations between these two sequences of digits?

The context is a business application: a generic number guessing game played with real money. If I use sequences that are cross-correlated, the player can leverage this fact (if she discovers the auto-correlations) to increase her odds of winning, making the game unfair to the operator. In short, I could lose money. For details, see section 4 in my article Some Fun with Gentle Chaos, the Golden Ratio, and Stochastic Number Theory.

## If a circle with a diameter of 12 contains a chord of length 6 sqrt2, what is the length of the minor arc intercepted by the chord?

I’ve been stuck on this question for a while now… the main problem is that I don’t know the way I should draw it.

Should I draw it so that the diameter is perpendicular to the chord (so it can bisect the chord & its arc), or should I create an inscribed angle with the chord and diameter?

I tried both ways and I still have no idea how to even get an answer. Any help is much appreciated!