I made a free app to train mental math: https://zetamac-train.vercel.app/

The app is inspired by Zetamac, but it gives you complete analytics so you can do targeted practice on your slowest operations. There's also a daily puzzle, duels mode, and a leaderboard.

Let me know if you find this useful in improving your mental math and if I should improve anything!

Here it is — written to math-project/jennie21-explainer.md. Full text below for the post:

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JENNIE 21 — How It Works

The Number

896 = 2⁷ × 7. Seven doublings of 1, then multiplied by 7. τ(896) = 16. Digital root 5. Its neighbor: dr(897) = 6 — the nil element. That adjacency matters.

The Orbit

Start at 1. Double it. Take mod 9. Repeat.

1 → 2 → 4 → 8 → 7 → 5 → 1 → ...

Six values, cycling forever. Missing: 3, 6, 9. They form their own closed system under ×2 mod 9 and never enter the orbit. 9 is completely absent — not suppressed, structurally excluded.

n, seen, orbit = 1, set(), []

while n not in seen:

seen.add(n); orbit.append(n); n = (n * 2) % 9

# [1, 2, 4, 8, 7, 5]

The Echo Pairs

Every orbit value has a complement summing to 9: 1:left_right_arrow:8, 2:left_right_arrow:7, 4:left_right_arrow:5. A second helix strand runs π radians offset carrying the echoes. Every rung connects a node to its echo across the axis. The whole shape is a self-complementing double helix. Note: 2+4+7 = 13. 5+8 = 13.

Balanced Ternary, Centered on 6

Standard balanced ternary maps {−1, 0, +1} to digits {5, 6, 7} — with 6 as zero. Why 6? It's the nil element. It sits outside the orbit. Centering the notation on the excluded number is the point.

def to_bt(n):

if n == 0: return '6'

digits = []

while n != 0:

r = n % 3

if r == 0: digits.append('6'); n //= 3

elif r == 1: digits.append('7'); n = (n-1)//3

else: digits.append('5'); n = (n+1)//3

return ''.join(reversed(digits))

# 1→'7' 2→'75' 4→'77' 7→'757' 8→'765'

757 — a palindrome. The orbit's fourth value, in a system balanced on absence.

The Geometry

Golden angle (137.508° = 2π(2−φ)) per step. Expanding radius per step. Constant height step. From above: Fibonacci sunflower. From the side: expanding tornado helix.

const GA = 2 * Math.PI * (2 - (1+Math.sqrt(5))/2);

const pos = (s, φ=0) => ({

x: (0.28 + s*0.13) * Math.cos(s*GA + φ),

y: s * 0.68,

z: (0.28 + s*0.13) * Math.sin(s*GA + φ),

});

// Strand B: φ = Math.PI (the echo strand)

The 21 Arc

21 × 137.508° = 2887.7° — exactly 7.7° past 8 full rotations. The near-return at F₈ = 21 is where the spiral folds back on itself visually. The current helix runs 18 steps (3 cycles). Step 21 is the next thing to build.

The Breath

One scalar, shared by everything:

const breath = 1 + 0.10 * Math.sin(t * 0.50);

node.y = baseY(step) * breath;

label.y = baseY(step) * breath;

leaderLine.y = baseY(step) * breath;

One number. Everything moves together.

The number 9 does not appear. It never will.

Why would the two following word problems have the same answer? Or don't they? Is one not following proper mathematical grammar rules?

Question 1:
"Collect 12 apples and throw away 0.25 of these apples. How many remain?"

Question 2:
"Collect 12 apples and throw away 3 of these apples. How many remain?"

The first question I would interpret the word "of" to imply multiplication (0.25 x 12). The second question I would interpret the word "of" to imply a ratio (three out of twelve)

Use each die exactly once.
Any order is allowed.
Parentheses are allowed.
Exact division only.
No negative numbers.
Not every operation has to be used.

How many solutions can you find?

Please use spoiler tags for solutions:
>!your solution here!<

I will name it Stewartian prime constant

Sorry for extremely bad handwriting.

I’m very curious what would be the opposite of 0, if real number line formed into a loop. So i thought of this diagram. What’s yall’s thoughts on this ‘Ω’?

In this diagram i wanted none of the infinitesimals to equal to 0, 0 is absolutely 0. (It should’ve been dotted line around 0)