From 8383481394533039d6087f626ce3505309bea2f4 Mon Sep 17 00:00:00 2001 From: rzuckerm Date: Sun, 18 Jan 2026 09:48:40 -0600 Subject: [PATCH 1/2] Add Zeckendorf in Every Language Article --- sources/projects/zeckendorf/description.md | 60 ++++++++++++++++++ .../projects/zeckendorf/featured-image.png | Bin 0 -> 33139 bytes sources/projects/zeckendorf/requirements.md | 8 +++ 3 files changed, 68 insertions(+) create mode 100644 sources/projects/zeckendorf/description.md create mode 100755 sources/projects/zeckendorf/featured-image.png create mode 100644 sources/projects/zeckendorf/requirements.md diff --git a/sources/projects/zeckendorf/description.md b/sources/projects/zeckendorf/description.md new file mode 100644 index 000000000..f76a3b6cb --- /dev/null +++ b/sources/projects/zeckendorf/description.md @@ -0,0 +1,60 @@ +[Edouard Zeckendorf][3] was a Belgian amateur mathematician who proposed a +[Theorem][1] that states that every non-negative integer (_N_) can be +represented as a sum of distinct non-consecutive [Fibonacci Numbers][2]. In +other words, + +* _N_ = _F__c_1 + _F__c_2 + ... + _F__c__n_ + +where: + +* _F__i_ is the _i_th Fibonacci number +* _c__i_ ≥ 2 +* _c__i_ < _c__i_ + 1 + +Each Fibonacci number (_F__i_) is the sum of the previous two Fibonacci +numbers: + +* _F_0 = 0 +* _F_1 = 1 +* _F_2 = _F_1 + _F_0 = 0 + 1 = 1 +* _F_3 = _F_2 + _F_1 = 1 + 1 = 2 +* ... +* _F__i_ = _F__i_ - 1 + _F__i_ - 2 + +This relationship is the reason why non-consecutive Fibonacci numbers are used. + +To find the Fibonacci numbers that add up to _N_: + +* Find all the Fibonacci numbers up to and including _N_, starting at + _F_2. +* Going from largest to smallest, select each Fibonacci number such that the + sum of all the selected values is less than or equal to _N_. + +For example, for _N_ = 67, these are the Fibonacci numbers up to 67: + +* 1, 2, 3, 5, 8, 13, 21, 34, 55 + +Therefore: + +| :Trial Sum: | :Fibonacci Number: | Selected | +| 0 | 55 | Yes | +| 55 | 34 | No | +| 55 | 21 | No | +| 55 | 13 | No | +| 55 | 8 | Yes | +| 63 | 5 | No | +| 63 | 3 | Ye | +| 66 | 2 | No | +| 66 | 1 | Yes | + +Therefore, using the selected numbers, _N_ can be represented as: + +* 67 = 55 + 8 + 3 + 1 + +A possible optimization for this proceedure would be to skip Fibonacci +numbers that are consecutive to the last selected value and to terminate +the loop as soon as the sum is equal to _N_. + +[1]: https://en.wikipedia.org/wiki/Zeckendorf%27s_theorem +[2]: https://en.wikipedia.org/wiki/Fibonacci_sequence +[3]: https://en.wikipedia.org/wiki/Edouard_Zeckendorf diff --git a/sources/projects/zeckendorf/featured-image.png b/sources/projects/zeckendorf/featured-image.png new file mode 100755 index 0000000000000000000000000000000000000000..9d9810e25c4b4ca9993cacaa72e09332a75748e4 GIT binary patch literal 33139 zcmZ6z1ymJn+cr#hNO!k1NVjxLcXxMp2}p-@OGtxAN=Qq0Nq2YmHyiKgUH|%VF?-Fp z=bG#6Ul?Er08XGR zMC3)lz-r?VpNyb^b5av2WqB|#Z)z~Gz)&!-2jEiR9vGM#3mDjeAs85M8WT;FwJDpFF4}+87|-=tgE!VIP3v9B04mUdObof 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+your language of choice. + +Write a sample program that accepts a single non-negative integer from the +command line and outputs a comma-separated list of Fibonacci numbers whose +sum equals the specified value. If the command line argument is missing or +not a non-negative integer, output the error message specified in the +[Testing](#testing) section. From 0736fd91bc27898616b8499690bceeddc885afbf Mon Sep 17 00:00:00 2001 From: rzuckerm Date: Sun, 18 Jan 2026 10:54:59 -0600 Subject: [PATCH 2/2] Fix typos --- sources/projects/zeckendorf/description.md | 38 +++++++++--------- .../projects/zeckendorf/featured-image.png | Bin 33139 -> 33252 bytes sources/projects/zeckendorf/requirements.md | 6 +-- 3 files changed, 23 insertions(+), 21 deletions(-) diff --git a/sources/projects/zeckendorf/description.md b/sources/projects/zeckendorf/description.md index f76a3b6cb..e02bab55e 100644 --- a/sources/projects/zeckendorf/description.md +++ b/sources/projects/zeckendorf/description.md @@ -7,11 +7,11 @@ other words, where: -* _F__i_ is the _i_th Fibonacci number +* _F__k_ is the _k_ th Fibonacci number * _c__i_ ≥ 2 -* _c__i_ < _c__i_ + 1 +* _c__i_ + 1 > _c__i_ + 1 -Each Fibonacci number (_F__i_) is the sum of the previous two Fibonacci +Each Fibonacci number (_F__k_) is the sum of the previous two Fibonacci numbers: * _F_0 = 0 @@ -19,7 +19,7 @@ numbers: * _F_2 = _F_1 + _F_0 = 0 + 1 = 1 * _F_3 = _F_2 + _F_1 = 1 + 1 = 2 * ... -* _F__i_ = _F__i_ - 1 + _F__i_ - 2 +* _F__k_ = _F__k_ - 1 + _F__k_ - 2 This relationship is the reason why non-consecutive Fibonacci numbers are used. @@ -28,32 +28,34 @@ To find the Fibonacci numbers that add up to _N_: * Find all the Fibonacci numbers up to and including _N_, starting at _F_2. * Going from largest to smallest, select each Fibonacci number such that the - sum of all the selected values is less than or equal to _N_. + sum of this value plus all the prior selected values is less than or equal to + _N_. For example, for _N_ = 67, these are the Fibonacci numbers up to 67: * 1, 2, 3, 5, 8, 13, 21, 34, 55 -Therefore: +Select the appropriate Fibonacci numbers: -| :Trial Sum: | :Fibonacci Number: | Selected | -| 0 | 55 | Yes | -| 55 | 34 | No | -| 55 | 21 | No | -| 55 | 13 | No | -| 55 | 8 | Yes | -| 63 | 5 | No | -| 63 | 3 | Ye | -| 66 | 2 | No | -| 66 | 1 | Yes | +| Trial Sum | Fibonacci Number | Selected | +| :---------: | :--------------: | :------: | +| 0 | 55 | ✅ | +| 55 | 34 | | +| 55 | 21 | | +| 55 | 13 | | +| 55 | 8 | ✅ | +| 63 | 5 | | +| 63 | 3 | ✅ | +| 66 | 2 | | +| 66 | 1 | ✅ | Therefore, using the selected numbers, _N_ can be represented as: * 67 = 55 + 8 + 3 + 1 -A possible optimization for this proceedure would be to skip Fibonacci +A possible optimization for this procedure would be to skip Fibonacci numbers that are consecutive to the last selected value and to terminate -the loop as soon as the sum is equal to _N_. +the loop as soon as the sum equals to _N_. 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a/sources/projects/zeckendorf/requirements.md b/sources/projects/zeckendorf/requirements.md index 66fd7f42d..b872dfef2 100644 --- a/sources/projects/zeckendorf/requirements.md +++ b/sources/projects/zeckendorf/requirements.md @@ -3,6 +3,6 @@ your language of choice. Write a sample program that accepts a single non-negative integer from the command line and outputs a comma-separated list of Fibonacci numbers whose -sum equals the specified value. If the command line argument is missing or -not a non-negative integer, output the error message specified in the -[Testing](#testing) section. +sum equals the specified value. For zero, output an empty list. If the command +line argument is missing or not a non-negative integer, output the error +message specified in the [Testing](#testing) section.