---
title: Four Small Tail-Merging Puzzles, Worked Through
url: https://share.jotbird.com/warm-vibrant-desert-raven
updated_at: 2026-10-25T15:08:06.561701+00:00
---

# Four Small Tail-Merging Puzzles, Worked Through

A tail-merging game can feel fast, but its number rule is small enough to study without a moving board. These examples use one convention: read the sequence from the inner body toward the outer tail, append the pickup at the right, then repeatedly combine only the final equal pair.

## 1. One useful match

Start: **8 → 4 → 2**  
Pickup: **2**

The last two 2s become 4. The new 4 matches the previous 4, producing 8; that matches the first 8, producing 16.

Final tail: **16**. Three merges occur after one pickup.

## 2. Same values, different order

Start: **8 → 2 → 4**  
Pickup: **2**

The ending pair is 4 and 2. It does not match, so the process stops immediately. The equal 2s farther inside the sequence do not jump over the 4.

Final tail: **8 → 2 → 4 → 2**. The total value is the same sixteen as in the first example, but the tail uses four segments rather than one.

## 3. A larger pickup is not always a better pickup

Start: **16 → 8 → 4 → 2**  
Pickup: **4**

The last pair is 2 and 4, so there is no merge. Compare that with picking up a 2: four successive matches would collapse the sequence into 32. The smaller tile is the one that completes the prepared chain.

This does not mean “always choose the smallest number.” It means “choose a number that fits the current end.” A different tail requires a different pickup.

## 4. A chain that stops partway

Start: **16 → 4 → 2**  
Pickup: **2**

Two 2s combine into 4, and the final two 4s become 8. Now the last pair is 16 and 8, which does not match.

Final tail: **16 → 8**. Two merges occur, and the tail is one segment shorter than it was before eating.

## Check your own prediction

There are two useful invariants. A merge preserves the sum of all segment values, and it decreases the segment count by exactly one. After appending a pickup and making m merges, the new length is the old length plus one minus m.

Try **32 → 16 → 8 → 4 → 2**, then append a 2. Predict the final sequence before checking the pairs. The answer is a single 64 after five merges.

## Add motion only after the rule is clear

The [Mergecoil puzzle at Snake 2048](https://snake2048.io/) puts this tail rule on a moving grid. It introduces a separate question: can the head reach a useful pickup and still escape safely? The arithmetic predicts a chain; it does not guarantee a safe route. Choose the Mergecoil puzzle for these examples, since the site's Arena game has a different loop.