HexaglyphHexaglyph Abacus
Learn to read Hexaglyphs (hex for hexadecimal, glyph for a written sign: one byte in one small shape), then calculate with them by moving marks instead of memorising tables, and finally meet signed digits, where hollow marks count negative and adding never needs a long chain of carries.
Reading a Hexaglyph
A Hexaglyph holds one byte, two hexadecimal digits, hung on a short bar called the stave. The high digit sits above the stave, the low digit below. Each digit is built from four marks that add up, like coins:
- 1 the left side 2 the right side 4 the outer edge 8 a dot in the middle
- So 0 is nothing, 3 is both sides, 7 is an open box, F is a closed box with a dot.
- The two halves mirror each other across the stave. Read the top as the first hex digit, the bottom as the second.
For text, each character is its keyboard code: digits have 3 above (both sides), capitals A–O have 4, P–Z have 5, lowercase a–o 6 and p–z 7. A capital and its lowercase differ by one stroke.
Build a glyph
Quick quiz
The abacus: arithmetic with marks
Every mark is worth twice the one before it. The ranks climb left side → right side → outer edge → dot, then cross the stave to the left side above it, then on into the next glyph. So arithmetic becomes a few rules:
- Add: pile the marks together; two of a kind make one of the next. Two dots below the stave carry across it; two dots above carry into the next glyph.
- Subtract: take marks away; if one is missing, break a bigger mark into two of the one below (borrowing).
- Double / halve: slide every mark up or down one rank.
- Multiply: keep doubling A; each mark of B picks a doubling to keep; add the kept rows (the Egyptian way).
- Divide: double B until it would pass A; take away the biggest doublings that fit; each gives the quotient a mark.
Signed digits: marks that count negative
Allow a second kind of mark: a hollow mark counts minus its value, while a solid mark counts plus. Every rank now holds −1, 0 or +1. Computer designers call this signed-digit or redundant binary, and it changes how arithmetic works.
One number, several spellings
A run of marks can always be replaced by one mark just past its top and one hollow mark at its bottom, because 0111 = 1000 − 0001. That gives every number a fewest-marks spelling, with no two marks on neighbouring ranks:
Why it’s worth it
- Adding without carry chains. In plain binary a carry can ripple all the way from one end to the other (FFFF + 1). With signed digits each rank only looks at its right-hand neighbour, decides what to keep and what to pass on, and the answer settles in two moves however long the numbers are. This is how some fast computer adders work.
- Subtracting without borrowing. Flip every mark of the second number (solid ↔ hollow) and add.
- Multiplying with fewer rows. Each mark of the multiplier is one row to add, and a hollow mark is a row to subtract. FF has 8 marks plain but only 2 in fewest-marks form, so A × FF is A × 100 − A.
- The price: a number has many spellings, so to read it plainly you subtract the hollow marks from the solid ones once, at the end.
Build a signed glyph
Tap a mark to cycle it: none → solid (+) → hollow (−).
Signed-digit abacus
Practice
Nine levels, from reading glyphs to adding signed digits. Pass a level with 8 of the last 10 right; your progress stays on this device. Answers are hexadecimal; put a minus in front for negative numbers.
Hexaglyphs grew from the Cistercian monks’ 13th-century numerals and the Reformed Cistercian Hexadecimal Numerals shared on Reddit. Signed-digit arithmetic: A. Avizienis (1961) and N. Takagi’s redundant binary adder (1985). Multiplying by doubling: the Egyptian scribes.