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Many know the TV game Countdown, whose French version Des chiffres et des lettres has lasted since 1965.

The rules of the count are as follows: you are given natural integers $n_1,\ldots,n_6$ and a target $N$. You are free to employ the four operations $+,\times,-,\div$. You may employ each $n_j$ at most once. You must end with the result $N$.

For mathematicians, a colleague of mine suggests to modify the rule that way: you are given $k\ge1$. You are free to choose $n_1,\ldots,n_k$. Then you must realize the targets $1,2\ldots,N$. How do you choose $n_1,\ldots,n_k$. What is the largest possible $N_k$ ?

Examples:

• $k=1$, nothing much interesting, $N_1=1$
• $k=2$, then $(1,3)$ yields $N_2=4$
• $k=3$, then $(2,3,10)$ yields $N_3=17$. Optimal ?

Edit about the rules. Parentheses are allowed (and useful). Division $a/b$ is possible only when $b$ divides $a$ in the usual sense of integers. You may have negative integers, but it does not help.

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Optimal Countdown

Many know the TV game Countdown, whose French version Des chiffres et des lettres has lasted since 1965.

The rules of the count are as follows: you are given natural integers $n_1,\ldots,n_6$ and a target $N$. You are free to employ the four operations $+,\times,-,\div$. You may employ each $n_j$ at most once. You must end with the result $N$.

For mathematicians, a colleague of mine suggests to modify the rule that way: you are given $k\ge1$. You are free to choose $n_1,\ldots,n_k$. Then you must realize the targets $1,2\ldots,N$. How do you choose $n_1,\ldots,n_k$. What is the largest possible $N_k$ ?

Examples:

• $k=1$, nothing much interesting, $N_1=1$
• $k=2$, then $(1,3)$ yields $N_2=4$
• $k=3$, then $(2,3,10)$ yields $N_3=17$. Optimal ?