What is it?

FHSS spreads a signal across many frequencies by hopping between them in a pseudo-random sequence. The transmitter and receiver agree on the sequence (a shared secret). A jammer who doesn't know the sequence can only block a fraction of the signal — the rest gets through.

The invention has an amazing origin story: Hedy Lamarr, a Hollywood actress, and George Antheil, a composer, patented frequency hopping in 1942 as a way to guide torpedoes without enemy jamming. The idea was decades ahead of its time — the US Navy only adopted it in the 1960s, during the Cuban Missile Crisis. Today it's everywhere: Bluetooth, WiFi (802.11), GPS, and military datalinks like Link 16 and SINCGARS.

Concrete Example

Suppose the answer word is HOP (3 characters). Each character is sent multiple times on different frequencies:

LCG: x_i+1 = (1103515245 · x_i + 12345) mod 2^31,  seed = 42

Hop schedule for HOP × 2 repetitions:
hop 0:  freq=17 → H (not jammed) ✓
hop 1:  freq=23 → O (jammed)      ✗
hop 2:  freq=4  → P (not jammed)  ✓
hop 3:  freq=12 → H (not jammed)  ✓
hop 4:  freq=23 → O (jammed)      ✗  (same channel, still jammed)
hop 5:  freq=9  → P (not jammed)  ✓

Char 0 (H) received at hops 0, 3 → both succeeded → H
Char 1 (O) received at hops 1, 4 → both jammed  → need 3rd redundancy
Char 2 (P) received at hops 2, 5 → both succeeded → P

The agent simulates the LCG from the seed, checks each hop against jammed channels, groups surviving transmissions by char_idx (hop_idx % N), and assembles the word.

How It Works

1. Synchronization: Transmitter and receiver share the same LCG seed. They hop together.
2. Hopping: At each time slot, both switch to frequency f = LCG(slot) % N.
3. Jamming: A jammer broadcasts noise on a subset of frequencies. If the hop lands on a jammed channel, the symbol is lost.
4. Redundancy: Each character is transmitted R times on different hops. If one hop is jammed, the others may survive.
5. Recovery: The receiver groups surviving transmissions by character index, fills gaps from the redundant copies, and reassembles the word.

Solving The Puzzle

def solve_fhss(data):
    # Simulate LCG to verify hop sequence
    a, c, m = data["lcg_a"], data["lcg_c"], data["lcg_m"]
    jammed = set(data["jammed_channels"])
    n = data["n_chars"]

    # Group surviving chars by character index
    chars = [None] * n
    for t in data["transmissions"]:
        if t["jammed"]:
            continue  # lost to jamming
        idx = t["char_idx"]
        chars[idx] = t["char"]  # any surviving copy works

    return "".join(chars).lower()

Difficulty Table

Real-World Applications

• Bluetooth: Jumps across 79 channels in the 2.4 GHz band, 1,600 hops/second
• WiFi 802.11: Uses FHSS in the original standard; modern WiFi uses DSSS/OFDM but FHSS lives in Bluetooth coexistence
• Link 16: NATO tactical datalink, 51 frequencies in the L-band, 77,000 hops/second
• SINCGARS: US Army single-channel ground-air radio system with FHSS
• GPS: Military P(Y) code uses a secret hopping pattern on L1/L2 frequencies
• Torpedo guidance: The original application — Lamarr and Antheil's 1942 patent for radio-controlled torpedoes using frequency hopping