In this paper, we show the explicit formula of the recurrence relation for the Tower of Hanoi on the star graph with four vertices, where the perfect tower of disks on a leaf vertex is transferred to the central vertex. This gives the solution to the problem posed at the 17th International Conference on Fibonacci Numbers and Their Applications[11]. Then, the recurrence relation are generalized to include the ones for the original 4-peg Tower of Hanoi and the Star Tower of Hanoi of transferring the tower from a leaf to another.

A Manhattan tower is a monotone orthogonal polyhedron lying in the halfspace z ≥ 0 such that (i) its intersection with the xy-plane is a simply connected orthogonal polygon, and (ii) the horizontal cross section at higher levels is nested in that for lower levels. Here, a monotone polyhedron meets each vertical line in a single segment or not at all. We study the computational complexity of finding the minimum number of guards which can observe the side and upper surfaces of a Manhattan tower. It is shown that the vertex-guarding, edge-guarding, and face-guarding problems for Manhattan towers are NP-hard.

A lot of improvements and optimizations for the hardware implementation of SubBytes of Rijndael, in detail inversion in F28 have been reported. Instead of the Rijndael original F28, it is known that its isomorphic tower field F((22)2)2 has a more efficient inversion. Then, some conversion matrices are also needed for connecting these isomorphic binary fields. According to the previous works, it is said that the number of 1's in the conversion matrices is preferred to be small; however, they have not focused on the Hamming weights of the row vectors of the matrices. It plays an important role for the calculation architecture, in detail critical path delays. This paper shows the existence of efficient conversion matrices whose row vectors all have the Hamming weights less than or equal to 4. They are introduced as quite rare cases. Then, it is pointed out that such efficient conversion matrices can connect the Rijndael original F28 to some less efficient inversions in F((22)2)2 but not to the most efficient ones. In order to overcome these inconveniences, this paper next proposes a technique called mixed bases. For the towerings, most of previous works have used several kinds of bases such as polynomial and normal bases in mixture. Different from them, this paper proposes another mixture of bases that contributes to the reduction of the critical path delay of SubBytes. Then, it is shown that the proposed mixture contributes to the efficiencies of not only inversion in F((22)2)2 but also conversion matrices between the isomorphic fields F28 and F((22)2)2.

In this paper, we analyze recurrence relations generalized from the Tower of Hanoi problem of the form T(n,α,β) = min 1 ≤ t ≤ n {αT(n-t,α,β) + βS(t,3)} , where S(t,3) = 2t - 1 is the optimal total number of moves for the 3-peg Tower of Hanoi problem. It is shown that when α and β are natural numbers, the sequence of differences of T(n,α,β)'s, i.e., {T(n,α,β) - T(n-1,α,β)}, consists of numbers of the form β 2i αj (i, j ≥ 0) lined in the increasing order.

The Towers of Hanoi problem is a classical problem in puzzles, games, mathematics, data structures, and algorithms. In this letter, a least memory used algorithm is proposed by combining the source array and target array for comparing the sizes of disk and labeling the disks in the towers of Hanoi problem. As a result, the proposed algorithm reduces the space needed from 2n+2 to n+5, where n represents the disks number.

This paper proves that for every positive integers n,k and any positive number ε, we can explicitly construct a DAG G with n+O(k1+ε) vertices and a constant degree such that even after removing any k vertices from G, the remaining digraph still contains an n-vertex dipath. This paper also proves that for every positive integers n,k and any positive number ε, we can explicitly construct a graph H with n+O(k2+ε) vertices and a constant degree such that even after removing any k vertices from H, the remaining graph still contains an n-vertex 2-dimensional square mesh.

The Tokyo Tower is the highest self-supporting steel tower in the world. Since it was built in 1958, the Tower has been a symbol of Tokyo and a well-known, major tourist attraction in Japan. The number of visitors reached 130 million in 1998. The highest number of visitors in one day was 40,000. The original purpose of the Tower was the transmitting of TV signals to the entire Tokyo Metropolitan area. As time passed, FM radio antennas and other equipment for public use were added to the Tower. Recently digital terrestrial antennas were installed on the Tower, a remarkable moment in its history. Digital broadcasting will start in 2003, using these antennas. This paper introduces the Tokyo Tower and its antennas, giving its construction history and its future in the coming digital broadcasting era.