This paper is uses a limited version of one section from of the more extensive units. If you have not solved some of these puzzles go to the puzzle page. This present paper merely illustrates a strategy which can be used solve all of his puzzles, altho with a limitation discussed later. This paper also suggests further considerations about Thinkadot puzzles analyzed in the more extensive units. If you are interested primarily in a mathematical analysis, you may want to skip this present paper.

The Thinkadot Device: Recall that this device has 3 holes at the top thru which the red marble can be dropped. It then falls thru gates inside the device, each set to send the marble left or right. A marble does not hit a gate in level 2 when dropped thru a side hole whose gate at level 1 is set toward the wall. Unless otherwise specified, it should be understood that we have started with the initial pattern in which all gates set to the left. As a marble goes thru a gate, it sets the gate in the opposite direction. On the outside in front of each gate is a blue dot if the gate is set left or a pink dot if it is set right. Let f, g, h denote the marble drops thru the left hole, middle hole, right hole respectively. The diagram below shows what you would see f followed by g followed by h.

The unseen inside gate settings are as follows.

Since any setting can be in one of 2 states, it is convenient to think in terms of bits rather than colors. A zero indicates that a gate is set so marble goes left. A one indicates that it is set so marble goes right. Thus you can think of the result of applying these marble drops in terms of the bit patterns below.

Notation: Using additive notation for marble drops, the above sequence of drops is denoted as f+g+h. Using multiples for repeated drops thru the same hole, g+2f means drop thru g then two drops thru f. You might want to verify that the bit pattern that results from applying the sequence 2h+ f+2g would be top 100, middle 10, bottom 010.

Strategy: We now describe a simple strategy for making any possible pattern x. It is based on the observation that we can obtain any bit pattern in level 1 using some subset of {f, g, h} and that further use of an even number of drops in a hole will not alter any bit setting in level 1, nor will 4 drops in a hole alter any bit setting in level 2.

Gate Change Details: Starting with any pattern, 2f changes the left 1^st level gate twice and the left 2^nd level gate once. Thus 4f changes these gates an even number of times. 4f also changes the left 3^rd level gate 3 times and the middle 3^rd level gate 3 once. Starting with any pattern, indicate the number of times each gate is changed by 4g. Do the same for 4h.

We suggest you use these observations about gate changes to solve some puzzles and try to discover and describe a general strategy before the example below.

Example: Four drops thru the same hole changes gates only in level 3. This observation can be used to make the pattern with level 1 as 111, level 2 as 00, level 3 as 010. This is sketched and explained below. To obtain the desired first level of 111 we need to change each bit, so we begin with f+g+ h. After this, the second level of 11 differs from the desired second level of 00 in both bits. 2g leaves level 1 alone, but reverses both bits in level 2. After f+g+h+2g the third level is 001, which differs from the desired bottom level 010 in its last 2 bits. 4h changes these 2 bits but leaves levels 1 and 2 alone.

Thus d = f+g+h+2g+4h will give pattern x.

Before reading our description of the general strategy you might try to apply a strategy like the above to some puzzles.

Top Down Strategy: The strategy just used for 111 00 010 can be used to find solve any of the puzzles given in Don Love's think-a-dot program. First use single drops to obtain the desired level 1. If level 2 needs to be changed then select an element of {2f, 2g, 2h}. If needed, select an element of {4f, 4g, 4h} to obtain the desired level 3. In more detail, let x be a pattern you are to obtain.

Limitations: To find a sequence of drops by the top down strategy, we first choose at most 3 drops, then at most 2 drops, then at most 4 drops; giving at most 9 marble drops. The pattern 111,00,010 is an example in which using this strategy involves for 9 drops, and it can be shown that this pattern cannot be obtained with fewer than 9 marble drops. However that there are patterns for which the top down strategy does not yield the minimal number of marble drops. This strategy is also limited because the marble drops to be used cannot be determined in advance. The Thinkadot Algebra unit uses mathematical concepts to find a strategy that will give the minimal number of drops without this limitation. The Thinkadot Group unit also gives a mathematical analysis that answers the question below and solves the following problems.

Question: Does the order in which drops are made affect to pattern?

Problem 1: Give a simple criterion for determining when a pattern is possible. For any possible pattern x, determine a sequence of marble drops that produces x. Find the minimum number of drops needed and a minimal sequence that produces x. Given any other natural number j determine if x can be obtained using a sequence of exactly j marble drops. If so, give such a sequence.

Problem 2: Let y and x be any patterns. Determine if there is a sequence of drops that changes y to x. If so, find such a sequence. Also find the smallest number s of drops needed to obtain x from y a sequence of length s that does this. Given any other natural number j determine if x can be obtained from y using a sequence of exactly j marble drops. If so, give such a sequence.

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