solidity
Perplexity on encoding deck of cards in uint256:
To encode a **full permutation** (the exact order) of a 52-card deck in Solidity using a `uint256`, you can use the **factorial number system** (also called Lehmer code). This system allows you to map any permutation of 52 elements to a unique integer between 0 and $$52! - 1$$, and vice versa[8][14].
### Key Points
- **Bits required:** $$ \log_2(52!) \approx 225.58 $$ bits, so a `uint256` (256 bits) is sufficient.
- **Encoding:** Convert the permutation to its Lehmer code, then to a single integer.
- **Decoding:** Convert the integer back to a Lehmer code, then reconstruct the permutation[14].
### How It Works
1. **Lehmer code:** For each position in the permutation, count how many remaining elements are less than the current element. This produces a sequence of 52 numbers, each less than or equal to its positional factorial base.
2. **Integer representation:** Each digit of the Lehmer code is multiplied by its factorial base and summed, resulting in a unique integer for each permutation[14].
### Solidity Implementation Outline
While the search results do not provide Solidity code for permutation encoding/decoding, the factorial number system is well-documented and can be implemented as follows:
#### Encoding a permutation to an integer (Lehmer code):
```solidity
function permutationToIndex(uint8[] memory perm) public pure returns (uint256 index) {
uint8[52] memory temp;
for (uint8 i = 0; i < 52; i++) temp[i] = i;
for (uint8 i = 0; i < 52; i++) {
uint8 v = perm[i];
uint8 less = 0;
for (uint8 j = 0; j < 52 - i; j++) {
if (temp[j] == v) {
index = index * (52 - i) + less;
// Remove used element
for (uint8 k = j; k < 51 - i; k++) temp[k] = temp[k+1];
break;
}
less++;
}
}
}
```
#### Decoding an integer to a permutation:
```solidity
function indexToPermutation(uint256 index) public pure returns (uint8[] memory perm) {
perm = new uint8[](52);
uint8[52] memory temp;
for (uint8 i = 0; i < 52; i++) temp[i] = i;
for (uint8 i = 0; i < 52; i++) {
uint256 f = factorial(52 - 1 - i);
uint8 pos = uint8(index / f);
perm[i] = temp[pos];
// Remove used element
for (uint8 k = pos; k < 51 - i; k++) temp[k] = temp[k+1];
index %= f;
}
}
```
(The `factorial` function must be implemented efficiently; precompute values if needed.)
### Summary Table
| Deck Representation | Bits Needed | Fits in uint256? | Encoding Method |
|---------------------------|-------------|------------------|------------------------|
| Full permutation (order) | ~226 | Yes | Factorial number system|
**References:**
- Factorial number system and permutation encoding[8][14].
This is the **most efficient way** to encode a full permutation of 52 cards in Solidity.
Sources
[1] Mathematical Operations in Solidity - GeeksforGeeks https://www.geeksforgeeks.org/solidity/mathematical-operations-in-solidity/
[2] Factorial contract errors in version 0.5.1 - Ethereum Stack Exchange https://ethereum.stackexchange.com/questions/72169/factorial-contract-errors-in-version-0-5-1
[3] EthIR/examples/factorial.sol at master - GitHub https://github.com/costa-group/EthIR/blob/master/examples/factorial.sol
[4] Calculate factorial | DApp World https://dapp-world.com/problem/math-calculate-factorial-tvfh/problem
[5] bc factorial - Posts - OneCompiler https://onecompiler.com/posts/3zz8q623f/bc-factorial
[6] Contracts — Solidity 0.8.31 documentation https://docs.soliditylang.org/en/latest/contracts.html
[7] Solidity Contract for demoing how to use libraries. The ... - GitHub Gist https://gist.github.com/ashwinYardi/029cfddd5b7eceb677d4b750cf064e68
[8] Factorial number system and enumerating permutations - Reddit https://www.reddit.com/r/math/comments/kz5sc/factorial_number_system_and_enumerating/
[9] Solidity by Example — Solidity 0.8.31 documentation https://docs.soliditylang.org/en/latest/solidity-by-example.html
[10] Solidity and Rust Syntax, Data Types, and Basic Constructs https://developers.stellar.org/docs/learn/migrate/evm/solidity-and-rust-basics
[11] Implementation of a nonlinear transformation of a substitution in the form of elements of a numbering system of a number of factorial sets https://bit.mephi.ru/index.php/bit/article/view/1365
[12] Design and implementation of NFT-based system https://drpress.org/ojs/index.php/HSET/article/view/4929
[13] IID-Based QPP-RNG: A Random Number Generator Utilizing Random Permutation Sorting Driven by System Jitter https://arxiv.org/abs/2502.18609
[14] Factorial Numbers and Their Practical Applications https://www.mdpi.com/2076-3417/14/19/8588
[15] A Universal Classification System of Skin Substitutes Inspired by Factorial Design. https://www.liebertpub.com/doi/10.1089/ten.teb.2017.0477
[16] Engagement is a necessary condition to test audit and feedback design features: results of a pragmatic, factorial, cluster-randomized trial with an embedded process evaluation https://implementationscience.biomedcentral.com/articles/10.1186/s13012-023-01271-6
[17] Improving medication adherence among persons with cardiovascular disease through m-health and community health worker-led interventions in Kerala; protocol for a type II effectiveness-implementation research-(SHRADDHA-ENDIRA) https://trialsjournal.biomedcentral.com/articles/10.1186/s13063-024-08244-0
[18] Implementation Intention and Reminder Effects on Behavior Change in a Mobile Health System: A Predictive Cognitive Model http://www.jmir.org/2017/11/e397/
[19] IMPACT OF DEMOGRAPHIC FACTORS GENDER AND NUMBER OF YEARS OF ERP POST IMPLEMENTATION OPERATIONS ON USER PRODUCTIVITY: AN EMPIRICAL STUDY IN HIGHER EDUCATION SETTINGS, IN INDIAN CONTEXT https://www.granthaalayahpublication.org/journals/index.php/granthaalayah/article/view/10_IJRG19_A11_2887
[20] PERFORMANCE COMPARISON OF THE LANGUAGE PROGRAMMING LUA AND C++ ON IMPLEMENTATION PROGRAM FIBONACCI AND FACTORIAL. https://www.semanticscholar.org/paper/e9e80120862824bc1d2de862f3ce35bfbed5cb77 0 reply
0 recast
2 reactions
0 reply
0 recast
3 reactions
1 reply
0 recast
3 reactions
1 reply
0 recast
1 reaction
1 reply
0 recast
5 reactions
0 reply
0 recast
0 reaction
2 replies
0 recast
4 reactions
0 reply
0 recast
4 reactions
0 reply
0 recast
0 reaction
0 reply
3 recasts
16 reactions
0 reply
0 recast
4 reactions
3 replies
0 recast
2 reactions
1 reply
0 recast
0 reaction
3 replies
1 recast
9 reactions
0 reply
0 recast
1 reaction
