Story

I was bored sitting in front of my computer and using a rectangle to select icons on my screen. I could select $1$, $2$, $3$, $4$, but not $5$ icons.

(Black squares are the icons. Note that it is possible to find rectangles with $1$, $2$, $3$ ad $4$ black squares in respectively.)

So I rearranged the icons into the following arrangement:

So in this arrangement, there exists a rectangle that includes $i$ black squares, where $1 \le i \le 6$. However, on the other hand, $6$ is not the maximum. For example, we can actually achieve $7$ with this arrangement.

Problem Formulation

$a$ is a $n$-iconic number if there exists an arrangement function $f:\{ x \in \mathbb{Z} | 1 \le x \le n\}^2 \rightarrow \{ 0, 1\}$, such that for $1 \le i \le a$, there exists a quadruple $(\alpha_i, \beta_i, \gamma_i, \delta_i)$ such that $$\sum_{j = \alpha_i}^{\beta_i} \sum_{k = \gamma_i}^{\delta_i} f(j, k) = i$$

Find the maximum $n$-iconic number.

If $a$ is an iconic number with an arrangement function $f$ with an additional criterion $$\sum_{j = 1}^n \sum_{k = 1}^n f(j, k) = a$$ then $a$ is $n$-perfect iconic

Is the maximum $n$-perfect iconic number the same as the maximum $n$-iconic number?

Find the maximum $n$-perfect iconic number.

SmallCases

For $n = 4$, the greatest iconic number and the greatest perfect iconic number is $12$, by the following construction:

5 Confusion in notation fixed

Story

I was bored sitting in front of my computer and using a rectangle to select icons on my screen. I could select $1$, $2$, $3$, $4$, but not $5$ icons.

(Black squares are the icons. Note that it is possible to find rectangles with $1$, $2$, $3$ ad $4$ black squares in respectively.)

So I rearranged the icons into the following arrangement:

So in this arrangement, there exists a rectangle that includes $i$ black squares, where $1 \le i \le 6$. However, on the other hand, $6$ is not the maximum. For example, we can actually achieve $7$ with this arrangement.

Problem Formulation

$a_n$ a$is a$n$-iconic number if there exists an arrangement function $f:\{ x \in \mathbb{Z} | 1 \le x \le n\}^2 \rightarrow \{ 0, 1\}$, such that for$1 \le i \le a_n$a$, there exists a quadruple $(\alpha_i, \beta_i, \gamma_i, \delta_i)$ such that $$\sum_{j = \alpha_i}^{\beta_i} \sum_{k = \gamma_i}^{\delta_i} f(j, k) = i$$

Find the maximum $n$-iconic number.

3 Fixed grammar