# Divide $40$ to $4$ parts such that every number from $1-40$ can be realized

How can you divide $40$ to $4$ parts such that every number from $1-40$ can be realized just by adding or subtracting those $4$ parts?

-

2
– Marc van Leeuwen Dec 13 '12 at 12:41

## 2 Answers

All integers from 1 to 40 can be expressed by adding or subtracting 1,3,9 and 27 in such a way that each number is used at most once, and it should be either added or subtracted.

-

You mean, "each number is used at most once". – TonyK Dec 13 '12 at 13:25
Oops. That's true. – Aneesh Karthik C Dec 29 '12 at 7:04

My answer is the same as Aneesh, but I'll try to give some informal intuition/justification of the answer. Think of it this way: What numbers would you have needed if you could only add them atmost once? So, you could say that the coefficients of each of these numbers would be either 0 or 1. This translates to the binary number system, and you would require 1, 2, 4, 8, 16, 32 to generate all numbers from 1 to 40 (all powers of 2 less than 40).

Now look at your question. Going on similar lines, the appropriate coefficients would be -1, 0 and 1. You can treat it as a base-3 (ternary) number system, and you would need all the powers of 3 (less than 40) to generate all numbers from 1-40, namely: 1, 3, 9 and 27.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9403878450393677, "perplexity_flag": "head"}