Making Every Number from 1 to N Under a Fixed Cycle of +, ×, −, ÷
Abstract
Start with the number $2$. At each move, combine two numbers already made, but the operations must be used in the fixed repeating order +,×,−,÷. We ask for the fewest moves needed to make every integer from $1$ to N. Since $2$ is already one of the numbers we want and each move makes at most one new number, at least N−1 moves are needed. We show that N−1 moves are also enough for every N≥ 9. Conventional induction cannot work, because a division that comes right after a completed interval {1,…,P} produces numbers already made. Instead we extend a completed interval {1,…,P} to {1,…,3P} all at once, which counting shows is the smallest multiplicative extension P→ kP that can work, and then adjust the last few moves to reach every other N. For 9≤ N≤ 33 we give explicit sequences, found by computer search.
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BibTeX
@misc{lesmana2026making,
title = {{Making Every Number from 1 to N Under a Fixed Cycle of $+$, $\times$, $-$, $÷$}},
author = {Sean Lesmana and Theodore Tjugiarto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10095},
eprint = {2610.10095},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10095},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Making Every Number from 1 to N Under a Fixed Cycle of +, ×, −, ÷
AU - Sean Lesmana
AU - Theodore Tjugiarto
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10095
UR - https://arxiv.org/abs/2610.10095
AB - Start with the number $2$. At each move, combine two numbers already made, but the operations must be used in the fixed repeating order +,×,−,÷. We ask for the fewest moves needed to make every integer from $1$ to N. Since $2$ is already one of the numbers we want and each move makes at most one new number, at least N−1 moves are needed. We show that N−1 moves are also enough for every N≥ 9. Conventional induction cannot work, because a division that comes right after a completed interval {1,…,P} produces numbers already made. Instead we extend a completed interval {1,…,P} to {1,…,3P} all at once, which counting shows is the smallest multiplicative extension P→ kP that can work, and then adjust the last few moves to reach every other N. For 9≤ N≤ 33 we give explicit sequences, found by computer search.
ER -