I met Tibor Gallai in 1929 [when he was known as Tibor Grünwald] when we were both in high school. We knew of each other's existence since we both worked at the 'Kozépiskolai Matematikai Lapok', a journal for high school students which appeared every month and published problems and their solutions by students. This periodical had an immense influence on Hungarian mathematics; many children before the age of 15 realised that they wanted to be mathematicians, and many of the well-known mathematicians as young people worked in this journal.

1. How many five-digit multiples of 3 end with the digit 6?
   
   
2. A straight line is drawn across an 8 ◊ 8 chessboard. It is said to pierce a square if it passes through an interior point of the square. At most how many of the 64 squares can this line pierce?
   
   
3. Inside an acute triangle $ABC$ is a point $P$ that is not the circumcentre. Prove that among the segments $AP, BP$ and $CP$, at least one is longer and at least one is shorter than the circumradius of $ABC$.

He was a cheerful, sparkling man. He loved company; he enjoyed telling anecdotes. With his sarcastic humour he could entertain his company superbly. He liked his colleagues and was an indispensable participant in the coffee-house meetings of mathematicians.

In the early 1930s Budapest, a group of young Jewish students regularly met in a park or took excursions to the countryside to discuss mathematics. ... the regular members of this remarkable group [were]: Paul (Pal) Erdős, Tibor Grünwald (later Gallai), Gergőr (Geza) Grünwald, Ester Klein, György (George) Szekeres, Lily Székely (later Sag), Paul (Pal) Turán, Endre Vázsonyi, and Marta Wachsberger (later Sved).

I expected this to be easy but to my great surprise and disappointment I could not find a proof. I told this problem to Gallai who very soon found an ingenious proof. L M Kelley noticed about 10 years later that the conjecture was not new. It was first stated by Sylvester in the 'Educational Times' in 1893. The first proof though is due to Gallai.

Paul Erdős told me that Tibor Gallai discovered the theorem of our prime interest in the late 1930s. He did not publish it either. It first appeared in a paper by Richard Rado (with credit to "Dr G Grünwald", which was Gallai's last name then; the initial "G" should have been "T" and must be a typo). Rado submitted this paper on 16 September 1939; it is listed in bibliographies as a 1943 publication, but in fact came out only in 1945; World War II affected all facets of life, and made no exception for the great Gallai result.

... is not only a first rate mathematician but also an excellent teacher. From 1945 to 1949 he taught in the Jewish high school for girls in Budapest. In one year he had 22 students. Six of them became mathematicians and one of them, Vera T Sós, became one of the leading mathematicians in Hungary.

[Gallai] taught me for four years from the age of fourteen. ... He gave some of us special tasks. He handed us the High School Mathematical Magazines, which only came as stencilled copies after the war. He sent us to competitions. Thanks to him, I was able to get to know Alfred Rényi, Rózsa Péter, and Paul Erdős when I was a high school student. But perhaps more importantly, it was Gallai who introduced me to the joy of understanding, of discovering, of the attractiveness of mathematics.

... was one of the most popular and successful teachers. In spite of the fact that he was fairly severe at the examination he was greatly beloved by his students. He and Rózsa Péter wrote a very interesting and excellent textbook of mathematics for high school students. He was always ready to help his colleagues and students when ever they needed advice or help in mathematical or personal matters.

The courses we have taken were Marxism-Leninism, this was a political course, the Mathematics, Geometry, Machine Sciences, Engineering drawing, Chemical Sciences, Russian Language and National Defence. The last one was a military course with the intent to prepare us to finish this course and become an officer of the Hungarian army. I personally enjoyed very much the technical courses especially mathematics with our professor Mr Tibor Gallai. The professors in a large room gave lectures with about two hundred students from various faculties. The individual tables and chairs were on a declining floor, so the visibility to the lecturer and the blackboard was excellent. Before the professor entered the room we were all standing as a respect for the professors. Later the details of the presentation and practice of the subjects was provided in a small room with about twenty students with the supervision of teachers. These were our classmates.

You may be surprised that I do not start with him as a mathematician, but rather about his human qualities. Gallai was an infinitely pure man. He stood for the highest of moral standards, morality, human integrity, of which there are very few examples. Even now, years after his death, he is often referred to at conferences: Gallai was the first to prove this point, but he never published it. There are several such items that have since been named after others. He raised his values high, published only his best achievements, his international influence is very significant. Here is a typical story: in 1956, the year of the Great Flood, he received the Kossuth Prize, but the money which came with the prize he gave, as it was, to the victims of the flood. Needless to say, Gallai was never a rich man. As usual, 'Matematikai Lapok' published an article about his award-winning work. Then Pál Turán, one of his best friends, was the editor-in-chief of the journal. Gallai wrote him an official letter in which he complained that the János Bolyai Mathematical Society journal had given such a commendable article about his work that he had, therefore, left the Society in protest.

Here is another image of the man. Gallai moved to the University of Technology in 1948-49, where he became head of the Department of Mathematics. He then resigned and headed to the Mathematical Research Institute. At the age of fifty-five, he left the Institute. He had saved little money, and the mathematician had nothing much to do at that time. For five years, he lived very poorly, in an ascetic manner, without pension or benefits. In fact, no one, not even his closest friends, have ever explained why he had left the Institute. Unfortunately, his wife soon became very ill. For more than ten years, Gallai had been with her from morning to night, doing everything required for such a disease. He had to stop doing mathematics. The exceptions were the short periods when Erdős was at home. He always went to Gallai and encouraged him to think about mathematics. When his wife died, Gallai was already seventy years old. He picked up the topics he once stopped and wrote some very serious articles. There remained a keen interest and passion for mathematics, even though he had overshadowed it for many years.

Gallai discovered a number of fabulous results, some of which were named after other mathematicians: he preferred not to publish even his greatest results. Why? On 20 July 1993 in Kesztely, Hungary, during a dinner my wife Maya, our baby Isabelle, and I shared with George Szekeres and Esther Klein, the legendary couple from the legendary circle of young Jewish mathematicians in early 1930s Budapest, I was able to ask them about the friend of their youth.

"Gallai was so terribly modest," explained George Szekeres. "He did not want to publish because it would show the world that he was clever, and he would be restless because of it.." "But he was very clever indeed," added Esther Klein-Szekeres. Esther continued: "Once I came to him and found him in bed. He said that he could not decide which foot to put down first." "Gallai was Paul Erdős's best, closest friend," continued George. "I was very close with Turán. It was later that Paul Erdős and I became friends."

Gallai believed in "communism" in spite of his reservations in the actual realisation. When I started to work for the Mathematical Institute (now the Renyi Institute) he was the party secretary there. Around 1967 or 1968 his wife became seriously ill. He decided to stay at home to be able to take care of her. But he was 2 or 3 years short of the retirement age. Therefore he had no income for these years. We said to him "you come to the Institute each week for an hour and work at home." He said "No, if I am an employee, I have to do some organisation work for the Institute and this needs more time to be there."