15.1 Letter Shifts, Reversal & Numerical Codes

Letter Shifts, Reversal & Numerical Codes

1. Letter Coding Transformations

Letter coding substitutes letters in a word according to a fixed geometric or numerical rule.

• Direct Positional Shift: Every letter shifts forward or backward by a constant $k$:
  • If $k = +3$: $\text{CABIN} \to \text{FDELQ}$ ($C+3=F, A+3=D, B+3=E\dots$)
• Cross-Pattern (Partition) Shifting: Words are bisected into symmetrical halves and reversed:
  • For an even 6-letter word divided into two 3-letter blocks $(1-2-3)$ and $(4-5-6)$:
    • Block 1 is reversed and shifted ($3-2-1$)
    • Block 2 is reversed and shifted ($6-5-4$)
    • Example: In a $+1$ cross-reversal of $\text{SIGNAL}$ (partitioned $\text{SIG} | \text{NAL}$):
      • $\text{SIG} \to \text{G-I-S} \xrightarrow{+1} \text{H-J-T}$
      • $\text{NAL} \to \text{L-A-N} \xrightarrow{+1} \text{M-B-O}$
      • Result: $\text{HJTMBO}$
• Alternating Reverse Substitution: Odd-positioned letters shift $+k$, while even-positioned letters are replaced by their reverse opposite letters ($27 - \text{Rank}$).

2. Numerical and Value Coding

Words are transformed into numerical values using alphabetical ranks:

• Positional Summation: Adding the forward alphabetical ranks of each letter:
  • $\text{CAB} = 3 + 1 + 2 = 6$
  • $\text{RAIL} = 18 + 1 + 9 + 12 = 40$
• Reverse Positional Summation: Adding backward ranks ($27 - \text{Rank}$):
  • For $\text{CAB}$: $(27-3) + (27-1) + (27-2) = 24 + 26 + 25 = 75$.
  • Shortcut: $\text{Reverse Sum} = 27 \times (\text{Number of Letters}) - (\text{Forward Sum})$.

$$\text{Reverse Sum}(\text{CAB}) = 27 \times 3 - 6 = 81 - 6 = 75$$

• Vowel/Consonant Partition Coding: Vowels are numbered by their sequence index ($A=1, E=2, I=3, O=4, U=5$) while consonants retain standard alphabetical ranks.

Use positions and modular wraparound

Coding is a rule-following task. Assign A = 1 through Z = 26 and check whether a transformation changes positions, reverses order, substitutes complements or combines operations. Record these operations separately. A code that happens to produce a familiar word is not evidence that the rule is correct.

For a forward shift by three, CABIN becomes FDELQ: C becomes F, A becomes D, B becomes E, I becomes L and N becomes Q. A shift must be applied to every position consistently. The incorrect string FDFLQ changes B by four and fails its own rule. For wraparound, X shifted forward by five becomes C, since the positions after X are Y, Z, A, B, C.

A general wrapped position is one plus the remainder after dividing old position minus one plus the shift by 26. The subtraction and addition of one preserve the one-based alphabet. Without them, a remainder of zero can be mistaken for a nonexistent letter instead of Z. For backward shifts, count cyclically or use a nonnegative remainder.

Distinguish reversal from complementary substitution

Reversing RAIL gives LIAR because the original letters change order. Complementing each letter gives IZRO because each original position p is replaced by 27 minus p. These are different operations. If a question says first reverse and then shift, carry out both in that order and test the supplied example.

For a compound illustration, split SIGNAL into SIG and NAL, reverse each block to GIS and LAN, then shift every letter forward by one. The result is HJT and MBO, producing HJTMBO. The middle boundary stays fixed throughout. Reversing the entire six-letter word would produce another output and would not follow the stipulated rule.

Numerical codes and sufficiency

When a word is coded as a number, test whether the rule uses letter sum, product, first/last positions or a stated weighting. CAB has rank sum 6, while its reverse-rank sum is 75: three times 27 minus 6. This does not imply that every three-letter word with sum 6 must be coded uniquely unless the rule is explicitly the sum. Different words can share the same numerical code.

An example alone can be underdetermined. CAB becoming FDE is consistent with a forward-three shift, but many contrived mappings could fit it. Use additional examples and the question's constraints to test the intended simple rule. Do not claim to have proved a unique encryption system from one short pair. In a timed item, consistency across all letters and examples is the practical verification standard.

Check an alternating rule

If odd-position letters shift forward by one and even-position letters shift backward by one, RAIL becomes SZJK: R to S, A to Z with wraparound, I to J and L to K. Apply the odd/even rule to positions, not to whether a letter is a vowel. Decoding reverses each position’s operation: the odd positions move backward one and even positions forward one. A successful round trip reconstructs RAIL and provides an independent check.

Test decoding without changing the rule

Under a forward-three code, ZOO becomes CRR: Z wraps through A and B to C, while each O becomes R. To decode CRR, move every position backward three. C returns to Z and each R returns to O. If a proposed decoding returns ARR, it has handled wraparound incorrectly or failed to reverse the operation.

Repeated input letters receive repeated output letters under a fixed position-independent shift. They need not do so under a position-dependent rule, such as alternating forward and backward shifts. Determine which kind of rule is supplied before treating repeated letters as an invariant.