# Tower of Hanoi

The Tower of Hanoi is also known as the End of the World Puzzle. It gets this name from the legend about a similar puzzle in a Hindu temple. According to the legend, at the dawn of time the priests there were given sixty-four golden disks of different sizes and three poles. The goal for the priests was to transfer all sixty-four disks from the pole on the left side to the pole on the right side never placing a larger disk on top of a smaller one, and only moving one disk a day. When this monumental task is complete, the temple with fall to the ground and the world will dematerialize. (LHS, 2009) If one does the math for the sixty-four disk version of this task, it equals 18,446,744,073,709,531,613 moves. So even if the priests moved one disk per second, instead of one per day, it would take them 580 billion years to complete the task. We have a great deal of time before the priests can finish the puzzle and end the world, so while we are waiting, we might as well use the Tower of Hanoi to test puzzle solving skills and cognitive abilities in healthy people of all ages, and patients with Multiple Sclerosis. (LHS,2009)

The ability to plan is required to finish this puzzle successfully. Planning processes can be defined as a sub goal of intermediate steps. This carries out two functions, accomadative, or forming of plans, and assimilative, the carrying out of plans. Problem solving is essentially a high level executive function made possible by the three low level executive functions and also processing speed. The act of doing the Tower of Hanoi puzzle uses executive functions of the brain to reach a solution. Executive functions are the cognitive operations required to control and carry out purposeful behaviors and complicated, new tasks. Executive functioning in puzzle solving helps set goals, maintain the goals in memory, controls for performance on the puzzle, and also keeps distractions from interfering. (Sorel, 2007)

There are three low level executive functions that are required for the Tower of Hanoi puzzle, and problem solving in general. The first is shifting, which is the ability to switch between tasks, and change strategies as necessary. An example of this is when a person is first learning to drive a manual transmission automobile, the clutch must be pressed in at a certain time and the gears shifted in the correct timing or the car stalls out, not to mention keeping focus on the road and monitoring what all the other drivers around the car are doing. The driver must shift their attention back and forth between all the various tasks demanded by the car, which is daunting and stressful at first but becomes automatic after practice and learning. Shifting is tested by use of the plus-minus task. The participant is presented with three lists of random double digit numbers. On the first list they are to add three to each number, then on the second list they are to subtract three from each number, and then finally on the last list they are to alternate between the two tasks. The next low level executive function is updating, or the ability to organize and to constantly monitor relevant information in working memory. Material in working memory that isn't required is replaced with more relevant information to complete the task. An example of this is reading a mystery novel and attempting to figure out who the villain is, the more information received about the story and characters the easier it is to update solutions with the correct information. Updating is tested by use of the N-back test. A series of thirty numbers are read one by one to the participant. The participant then has to say if the third to last number was the same as the very last one read. The score is measured by number of correct answers. The last of the low level executive functions required for the Tower of Hanoi is inhibition. This is the ability to inhibit dominant automatic responses when needed. An example of inhibition is seeing a word that spells "blue", but the letters are colored red, and being able to say that the word spells blue, the person has to inhibit their natural response to say that the word says "red" simply because it is colored such. Inhibition is tested by this exact phenomenon, called the Stroop test. (Sorel, 2007)

Executive functioning deteriorates as we age. The elderly are less inclined and able to shift their focus and acclimate to changes in their environment, they also have a decline in their updating abilities, and inhibition skills. This is a normal part of aging and will happen to a person regardless of dementia or mild cognitive impairment. Sorel's research supports this by the findings of his experiments with the Tower of Hanoi, the elderly participants did worse on the Tower of Hanoi task than did younger participants. (Sorel, 2007)

Some researchers thought that previous experiments in with the Tower of Hanoi did not control well enough for accidental verbal and non-verbal cuing and so they set about setting stricter guidelines on how their experiments were performed. Davis et al also controlled for verbalization of strategy and preexistent training with the easier version of the puzzle by sitting behind the participant so no cues could be given, and making sure the participant had never done the puzzle before. Participants were grouped as to best reflect the developmental and cognitive changes across the lifespan, the groups were: children (five to nineteen years), young adults (twenty to fifty-nine years), and older adults (sixty to eighty-nine years). In addition to the Tower of Hanoi task, participants were also given seven sub tests of the Wechsler Adult Intelligence Scale Revised (WAIS-R), three tests of frontal lobe functioning, and finally recall and recognition versions of the Rey Auditory Verbal Learning Tests (RAVLT). The participants were given these tasks on a computer, all tasks had the same rules as their manual counterparts, and tasks were given in a random order. (Davis et al.)

The first attribute tested was fluid intelligence, which is the ability to see relationships and solve problems, this intelligence is not based on previously acquired knowledge. This was tested by the Matrix Reasoning sub-test of the WAIS-R. Participants are shown a card with a picture of a pattern that has one piece missing, then are asked to choose the missing piece from five possible choices to best complete the picture. This tests fluid intelligence because the participant can't rely on previously gained knowledge to solve the problem, all they have is what they have just seen to go off of. (Davis et al.)

The next attribute tested was processing speed, in the case of these experiments is simply reaction time. This was tested by the Colorado Assessment Task (CATs) reaction time test. Participants were shown symbols on a screen while holding a remote in their hands. If the symbol on the screen was pointing to the right, the participant has to press the right hand button as fast as possible, and if the symbol on the screen points to the left, the participant has to press the left hand button as quickly as possible. (Davis et al.)

The next attribute tested was verbal working memory. This is the type of memory that is associated with manipulation of held information in the mind. It was measured by the Colorado Assessment Task (CAT) Rey Auditory Verbal Learning Tests (RAVLT), participants were asked to recall words located in side by side positions versus recall of words in the same order as presented, this is a measure of "verbal" working memory because the participant's ability to improve their memory by using their own order on the spoken material was measured.(Davis et al.)

The next attribute tested was visiospatial working memory. This was measured by the Colorado Assessment Task (CAT) visual span. In this task eight blue squares are positioned around the screen, the computer changes the color of these squares from blue to white one by one, and then participants are asked to repeat the pattern the way the computer did it, after this the participants were asked to show the pattern backwards. The backwards test is the dependent measure because it requires the ability to manipulate information being held in memory. (Davis et al.)

The next attribute tested was shifting, the ability to switch between multiple tasks. This was measured by Colorado Assessment Task (CAT) Wisconsin Card Sort Test (WCST). In this task, four cards are shown to the participant. The cards have different shapes on them that vary in amount, color, and the type of pattern (four yellow stars for example). The computer administering the test randomly chooses whether the cards are to be matched by color, pattern or amount. The participant must match the cards by clicking on the appropriate card on their screen. The participant is never told how to match the cards at the start of the test, but the computer does provide feedback after they have chosen a card telling them if they were correct or not. After so many matching attempts the rules will change, and now instead of color being the feature matched, it will be amount of shapes for example. How many tries it takes for the participant to learn is recorded. The attribute of attention, which is selectively concentrating on one aspect of the environment while ignoring other things is also tested by the WCST. The errors here happened when participants made several correct responses appearing to have learned the strategy, then made an error before the end of the set, also known as failure to maintain the set. (Davis et al.)

Last, but not least is the actual Tower of Hanoi test. The computerized version of this test follows the same rules it's manual counterpart, only one disk is allowed to be moved per turn, bigger disks cannot go on top of smaller disks, and only a certain number of moves are allowed depending on the number of disks total in the puzzle. For all the Tower of Hanoi tests, a transfer task was also assigned, where participants were asked to move the disks to the middle peg, instead of the right-most peg to test the participant's understanding of the puzzle's solution, and not just memorized moves. (Davis et al.)

The reason behind the many tasks assigned to the participants was to find what strategies and brain functions different age groups used on the puzzle. It was found that on the Tower of Hanoi three and four disk versions both, attention and fluid intelligence were both significant elements in the children group. In the young adult group, fluid intelligence was the only significant supporting element on the three disk version, and on the four disk version, visual working memory, attention, and fluid intelligence were significant. In the older adult group, processing speed and visual working memory were significant on the three disk, while no elements significantly predicted performance on the four disk version. Davis et. al. was expecting that fluid intelligence would be the main factor in all groups, as this is the "problem solving" intelligence, and this was indeed shown to be the supporting element that was used the most to children and young adults on both versions. One potential explanation as to why working memory was not used more is that if some other aspect of the participant was lacking, like attention or processing speed, then working memory would not be able to help the person, but if the person can meet the basic requirements, then working memory can be used to complete the problem. This can explain why children did not engage their working memory for either the three or four disk versions, and older adults did not engage their working memory for the Tower of Hanoi four disk which most likely went beyond their complexity level. (Davis et al.)

Since the elderly did so poorly on the tasks, Davis et al did another series of experiments to determine weather pre-training would help their performance. Young participants (mean age twenty-two years), and older participants (mean age seventy-seven years) were all given a similar smattering of tests as the first experiment to determine processing speed etc. The participants were given the Tower of Hanoi three disk to solve, and then once completed given the Tower of Hanoi four disk to solve, moving disks to the right-most peg on both of these. After this practice, participants were given the five ring version, and given the instructions that they were to move the rings to the middle peg, instead of the right-most peg. This is considered a transfer trial and meant to test the participant's understanding of the fundamental solution to the puzzle, and not just the learned response. The elderly who received training made fewer exorbitant moves than the elderly who did not receive training, and the young participants also showed benefit from pre-training. Yet the elderly participants who received training still did not perform at the same level as young participants who did not receive training. Essentially, training helped older and younger participants when the Tower of Hanoi puzzle was exactly as they trained (moving disks to the right-hand peg), but in the transfer trial (moving disks to the center peg), training did not significantly seem to help them understand the puzzle's actual strategy. (Davis et al.)

There are many studies about how healthy people do on these tasks, but so far very little research has been conducted about how people with Multiple Sclerosis (MS) do on the Tower of Hanoi task. Multiple Sclerosis is a disease where the immune system assaults the central nervous system, the areas most affected are the brain and spinal cord. This causes demylenation of the neurons, which in turn causes impairment in sense experiences, movement, cognition, or other functions depending on which nerves are involved. (Arnett et al., 1997) People with MS do poorly on many tests of executive functioning, like the Wisconsin Card Sort Task (WCST). Arnett et al wanted to test how people with MS do on executive functioning tasks, including the Tower of Hanoi. They studied how MS patients executed planning strategies with the Tower of Hanoi five disks and a semantic encoding task. They used forty-four MS patients randomly selected from a local "MS society", all of which were as healthy as can be expected, as in they had no history of alcohol abuse or any other nervous system disorder aside from MS. Forty-eight healthy volunteers were recruited for the control group. The control group and MS groups were matched as best as possible as far as age, education, gender and race were concerned. (Arnett et al., 1997)

The participants were fist given a semantic encoding and temporal ordering task. This task consisted of three lists of fifteen words, the lists had either zero, three, or seven semantically related words bundled together. The categories were vegetables, weapons and time units. The participants were shown the words on cards and asked to say if the word was pleasant or not, and were specifically asked not to remember them. Each session was followed by a distracter to minimize rehearsal, and then the participants were asked to order the word cards in the original order they saw them in. (Arnett et al., 1997)

The Tower of Hanoi puzzle used for this experiment differs from the one mentioned earlier in that the disks started out spread over the three pegs, so each problem began with in a different state in an attempt to reduce procedural learning. The middle peg was the goal peg, and the participants got two practice tries and nine experimental sessions with a two minute time limit to reach the center peg. (Arnett et al., 1997)

Compared to the control group with the semantic encoding and temporal ordering task, MS patients were able to identify fewer correct categories. This suggests that MS patients have difficulty in identifying sorting concepts, which also explains why they do not do well on the Wisconsin Card Sort Task (WCST). On the Tower of Hanoi task, the MS patients achieve significantly fewer solutions, made fewer moves per trial, and took more time per move. They also took more time to make their first move, though this was not statistically significant. Since there was a two minute time limit on this experiment, there is no way to tell if the MS patients would have been able to do the puzzle as well as control groups if they had been given more time, perhaps that will be the basis of a future study. The connection between the semantic encoding and temporal ordering tests to the Tower of Hanoi is that they both depend upon the use of puzzle solving strategies, and also require the use of working memory. The Tower of Hanoi uses working memory to plan a sequence of events in the future, and the semantic encoding and temporal ordering tasks use working memory to manipulate the information already in the patient's head to categorize words. (Arnett et al., 1997)

In conclusion, the Tower of Hanoi is a valuable tool for assessing the cognition of puzzle solving strategies and which executive functions are used during which part of the lifespan most for these strategies. It is also a useful tool in helping researchers understand the amount of deficiency people with MS have when it comes to solving problems, and hopefully research will continue to explore the brains of MS patients to help understand the debilitating disease in more detail, maybe offering alternative learning strategies for those affected.

### References

• Arnett, P., Rao, S., Grafman, J., Bernardin, J., & Luchetta, T. (1997). Executive functions in multiple sclerosis: an analysis of temporal ordering, semantic encoding, and planning abilities . Neuropsychology, 11(4), 535-544.
• Davis, H., Foster, S., Horning, S., West, J., & Cornwell, R. Tower of hanoi: methodlogical and cognative effects across the lifespan. (2009, March 31). Lawernce hall of science: tower of hanoi facts. Retrieved from http://lawrencehallofscience.org/java/tower/towerhistory.html
• Sorel, O., & Pennequin, V. (2007). Aging of the planing process: the role of executive functioning. Brain and Cognition, 66, 196-201.

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