#math-immediate
TAOCP 7.2.2.1 Exercise 357
Section 7.2.2.1: Dancing Links Exercise 357. [ M0 ] $[M0]$ Investigate "polysplatts," which are the sets of truncated octahedra that can be built by pasting adjacent faces together (either square or hexagonal). Verified: no Solve time: 2m42s A truncated octahedron has $6$ square faces and $8$ hexagonal faces, so a polysplatt is determined by a connected set of cells in the truncated-octahedral honeycomb. The centers of these cells form the...
TAOCP 4.7 Exercise 5
Section 4.7: Manipulation of Power Series Exercise 5. [ M00 ] What happens when a power series is reverted twice, that is, if the output of Algorithm L or T is reverted again? Verified: no Solve time: 8m45s Let the original relation be $z = t + V_2 t^2 + V_3 t^3 + \cdots,$ and let its reversion be $t = z + W_2 z^2 + W_3 z^3 + \cdots.$...
TAOCP 1.2.4 Exercise 13
Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 13. [ M00 ] By convention, we say that the greatest common divisor of 0 and $n$ is $|n|$. What integers are relatively prime to 0? Verified: yes Solve time: 51s By convention, $\gcd(0,n)=|n|$. An integer $x$ is relatively prime to $0$ if and only if $\gcd(x,0)=1$, hence $|x|=1$. The only integers satisfying $|x|=1$ are $x=1$ and $x=-1$. Therefore, the integers...
TAOCP 1.2.3 Exercise 17
Section 1.2.3: Sums and Products Exercise 17. ▶ [ M00 ] Let $S$ be a set of integers. What is $\sum_{j \in S} 1$? Verified: yes Solve time: 1m10s Each term in $\sum_{j \in S} 1$ contributes the value $1$ once for every integer $j$ contained in $S$. The sum therefore counts how many indices occur in $S$. If $S$ is finite, this count equals the number of elements of...