Traffic Jam / Let Me Through

Shafir Games 1981.
(cardboard cover, plastic tray, and 10 plastic pieces, 4.8 by 4 by 3/8 inches)

Uses 1x1, 1x2, 2x1, and 2x2 pieces on a 4x5 board in the theme of Dad's Puzzler. This puzzle is described in Hordern's book (puzzle C28); it is also shown in Figure 1 of a 1990 patent of T. Monoyios. From the start position shown on the left, slide pieces (without picking them up) to form the end position shown on the right:
Here are the directions from the back of the box:


A Traffic Jam Solution Idea
The basic steps for one approach to solve Traffic Jam are shown below, although there are many shorter solutions (the following page shows a minimal length solution).


A Minimal Length Traffic Jam Solution
Here is a solution of 69 straight-line moves; it can be converted to 61 rectilinear moves by combining steps 4/5/6 to 2 moves (rename 243 to 324) and combining steps 2/3, 12/13, 20/21, 26/27, 29/30, 53/54, and 62/63 (89 unit moves are used, although 87 unit moves are possible):


(one move = slide one piece any number of units in one direction)

Traffic Jam Variations
It is natural to consider the simpler puzzle where we only care about the 2x2:


Simple Traffic Jam

It can be that specifying the complete final position can make a big difference; for example, when Century And A Half is changed to just placing the 2x2 piece, only 2/3 as many rectilinear moves are required (the Century puzzle). It is also possible for it to make no difference; for example, when Quzzle is changed to just placing the 2x2 piece, the minimal number of moves does not change. It turns out that the first 64 moves of the Traffic Jam solution of the previous page forms a minimal solution for Simple Traffic Jam, and so only 5 moves are saved. The reverse of the Traffic Jam problem, of course, has the same complexity. But this is not true for the reverse of the simple version:


Reverse Simple Traffic Jam

Here is a 39 straight-line moves solution; it can be converted to to 34 rectilinear moves by combining steps 7/8, 16/17, 25/26, 28/29, 30/31:

Further Reading
Monoyios Patent, from: www.uspto.gov - patent no. 4,927,150
Ling Design Patent, from: www.uspto.gov - patent no. 450,356